Further author information: Send correspondence to F.X. Schmider
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# Observing exoplanets from Antarctica in two colours: Set-up and operation of
ASTEP+

François-Xavier Schmider  Université Côte d’Azur, Observatoire de la Côte
d’Azur, CNRS, Lagrange Laboratory, Nice, France  Lyu Abe  Université Côte
d’Azur, Observatoire de la Côte d’Azur, CNRS, Lagrange Laboratory, Nice,
France  Abdelkrim Agabi  Université Côte d’Azur, Observatoire de la Côte
d’Azur, CNRS, Lagrange Laboratory, Nice, France  Philippe Bendjoya  Université
Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Lagrange Laboratory, Nice,
France  Nicolas Crouzet  Leiden Observatory, Leiden University, Postbus 9513,
2300 RA Leiden, The Netherlands  European Space Agency (ESA), European Space
Research and Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The
Netherlands  Georgina Dransfield  School of Physics & Astronomy, University of
Birmingham, Edgbaston, B15 2TT, Birmingham, UK  Tristan Guillot  Université
Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Lagrange Laboratory, Nice,
France  Olivier Lai  Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Lagrange Laboratory, Nice, France  Djamel Mekarnia  Université Côte
d’Azur, Observatoire de la Côte d’Azur, CNRS, Lagrange Laboratory, Nice,
France  Olga Suarez  Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Lagrange Laboratory, Nice, France  Amaury H.M.J. Triaud  School of
Physics & Astronomy, University of Birmingham, Edgbaston, B15 2TT, Birmingham,
UK  Philippe Stee  Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Lagrange Laboratory, Nice, France  Maximilian N. Günther  European Space
Agency (ESA), European Space Research and Technology Centre (ESTEC),
Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands  Dennis Breeveld  European
Space Agency (ESA), European Space Research and Technology Centre (ESTEC),
Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands  Sander Blommaert  European
Space Agency (ESA), European Space Research and Technology Centre (ESTEC),
Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands

###### Abstract

On December 2021, a new camera box for two-colour simultaneous visible
photometry was successfully installed on the ASTEP telescope at the Concordia
station in Antarctica. The new focal box offers increased capabilities for the
ASTEP+ project. The opto-mechanical design of the camera was described in a
previous paper[1]. Here, we focus on the laboratory tests of each of the two
cameras, the low-temperature behaviour of the focal box in a thermal chamber,
the on-site installation and alignment of the new focal box on the telescope,
the measurement of the turbulence in the tube and the operation of the
telescope equipped with the new focal box. We also describe the data
acquisition and the telescope guiding procedure and provide a first assessment
of the performances reached during the first part of the 2022 observation
campaign. Observations of the WASP19 field, already observed previously with
ASTEP, demonstrates an improvement of the SNR by a factor 1.7, coherent with
an increased number of photon by a factor of 3. The throughput of the two
cameras is assessed both by calculation of the characteristics of the optics
and quantum efficiency of the cameras, and by direct observations on the sky.
We find that the ASTEP+ two-colour transmission curves (with a dichroic
separating the fluxes at 690nm) are similar to those of GAIA in the blue and
red channels, but with a lower transmission in the ASTEP+ red channel leading
to a 1.5 magnitude higher B-R value compared to the GAIA B-R value. With this
new setting, the ASTEP+ telescope will ensure the follow-up and the
characterization of a large number of exoplanetary transits in the coming
years in view of the future space missions JWST and Ariel.

###### keywords:

Exoplanets, transit, Antarctica, Concordia station, TESS, photometry

##  1 INTRODUCTION

ASTEP (Antarctic Search for Transiting ExoPlanets) has been developed to
search for exoplanetary transits from Dome C[2, 3, 4]. It has been able to
produce high-precision photometric observations, both for detection of new
candidates and for the follow-up of already known exoplanets[5, 6, 7, 8, 9,
10, 11, 12, 13, 14]. The location of ASTEP in Antarctica makes it unique for
the follow-up of long duration transits and very southern objects detected by
TESS and in the continuous viewing zones of JWST and Ariel. In 2020, a new
focal box was developed at the Lagrange laboratory with the support of
ESA/ESTEC and the University of Birmingham. The support permitted the
acquisition of two cameras for two-color (blue and red parts of the visible
light) simultaneous photometry. The focal box design and its characteristics
are described in a previous SPIE paper [1].

After the development and tests at the laboratory, the new focal box was
shipped and installed at the Concordia station, Dome C in Antarctica. The new
ASTEP+ telescope has started its observations in March 2022 for the whole
winter season.

Here we describe the different steps of the assembly and tests of the focal
box, its shipment to Dome C, its installation on ASTEP at Concordia during the
austral summer season, as well as its operation during the first months of the
polar winter season and the results achieved. In a parallel paper [15], we
describe the target selection process and the data automatic reduction, in
view of a fully robotic observation facility.

##  2 FOCAL BOX ASSEMBLY AND LABORATORY TESTS

Before shipping the focal box to the Concordia station, it as necessary to
assemble it at the laboratory and test its behaviour for the expected
environmental conditions in Antarctica, where the external temperature can
vary from -20∘C down to -80∘C. These conditions impose a special conception of
the box in different compartments with different functions, an external
isolation and internal temperature regulation to ensure the correct working
properties of the focal box.

###  2.1 Assembly

In the previous paper [1], we described the optical conception and the
realisation of the different mechanical supports for the optics and the
cameras. The main carbon composite plate and all opto-mechanical modules of
the new focal box were assembled on an optical bench in the laboratory to
align the optics. In addition to the mechanical assembly, several heating
elements were added in the different compartments of the focal box, so we
could guarantee different working temperature following the requirements of
each component. The optical compartment could remain colder than 0∘C, when the
cameras and their electronics have to stay in a positive temperature
environment. Thermal insulation was added all around the box and between each
compartment. The temperature control of the different components is carried
out by several stand-alone regulators with PID algorithm using PT100 probes.

---
Figure 1: The new focal box assembled in the lab for alignment with an
artificial optical source (right)

###  2.2 Laboratory tests

At the laboratory, several tests were carried out on each element before
assembly and set-up. Each camera was placed on an optical bench with a
monochromator and an integrating sphere (Figure 2) in order to estimate the
Pixel Response Non Uniformity (PRNU), the linearity of the detector and the
read-out noise level. The whole box was then aligned on an external light
source and the quality of the image was verified using an optical pattern
(Figure 3).

A first adjustment of the opto-mechanical modules consists of aligning all the
elements with a laser and, at the entrance of the box, of positioning a grid
located in a tube (telescope simulator) at the theoretical focal plane of the
telescope. The alignment of the optical axes of the two cameras with the
optical axis of the telescope is obtained by centering the image of the
pattern on each camera, which is done thanks to the adjustment of the dichroic
beamsplitter for the blue channel and of the mirror for the red channel.

---
Figure 2: FLI camera test behind the optical source
---
Figure 3: Optical pattern as seen by the FLI & ANDOR cameras

The optical transmission of the new focal box (Fig. 9) was estimated taking
into account the cameras QE, the transmission and reflection of the dichroïc
beamsplitter and the coating of the lenses and of the mirrors. The primary
(M1) and secondary (M2) mirrors of the telescope are coated with protected
aluminium, which has a gap in reflection around 800 nm, explaining the gap in
the transmission of the red channel. The central wavelength of the blue
channel is 555 nm, with a FWHM of 293 nm, while the central wavelength of the
red channel is 850 nm, with a FWHM of 276 nm. In the future, we foresee to
change the coating of M2 mirror, inclined at 45 degrees, by a protected silver
coating to increase the total transmission efficiency of the instrument. In
the section 4, we show the response of the focal box as compared to GAIA for
all measured stars in a given field.

The thermal behaviour of the box was tested in a thermal chamber. The external
temperature was varied to simulate the conditions at Dome C both in summer and
in winter, with a temperature range between -20∘C and -80∘C. The lower part of
the camera box contains electronic modules, in particular the Andor camera
power supply control electronics and a converter that allows the USB signal to
be transferred from the cameras to the acquisition computer. These electronic
devices produce heat when cameras are in function. This part of the box is
also thermalized using dedicated temperature controllers. Fans installed in
the compartment of each camera produces heat exchange between these
compartments and, thus, maintain the whole at a set temperature around 10∘C.
Tests in the thermal chamber were carried out which showed that 50% of the
heating power was necessary to guarantee the good functioning of the whole
installation in extreme cases and 80% during the periods when scientific
cameras are not dissipating heat (off mode).

---
Figure 4: Left: The new focal box is placed inside a thermal chamber, able to
reach a temperature of -75∘C, similar to that of the Antarctica winter. The
graphics on the right displays the temperatures measured at different places
inside the box obtained when varying the environment temperature.

##  3 SET-UP and OPERATIONS

###  3.1 Installation

The new box was shipped to Dome C on October 5th, 2021 and was installed on
the telescope during the austral summer campaign. The summer campaign was
organised between November 23rd and January 25th by the ASTEP team composed of
Karim Agabi, Djamel Mekarnia, Georgina Dransfield and Olivier Lai.

---
Figure 5: Aereal view of the Concordia station. The two main buildings of the
base are located in the center, while the summer camp is in the background of
the picture. The new location of ASTEP telescope is the white Baader dome
located in the foreground close to the Astro shelter.

During the summer campaign, we seized the opportunity to move the telescope to
a new location on the base, in the so-called Baader dome. This displacement,
already scheduled and prepared one year before, aims at grouping the French
and Italian astronomical activities in the same laboratory (called Astro
shelter). The new dome is better suited for the ASTEP+ telescope as it offers
more space and an automatic closure capability. In addition to this huge
operation, we also removed all ASTEP experiment elements from their initial
installation to their new location: electronics devices were installed close
to the new Dome and the telescope control computers were installed inside the
Astro shelter. We also installed a new primary mirror on the telescope, a new
data-processing server, conducted a whole maintenance of the telescope and its
mount and updated all scripts linked to the data acquisition and data
processing.

---
Figure 6: Left: Installation of the ASTEP+ telescope in its new dome, using
the logistical means of the station. Right: The ASTEP+ telescope equipped with
its new focal box fully installed in the new dome.

After the displacement of the telescope, the new box was placed on the
telescope. The optical alignment of the focal box was first checked at the
laboratory, using an internal light source, before its installation on the
telescope. Then, the alignment of the telescope with the focal box was done by
pointing a bright star, usually Canopus, visible in daylight, thanks to the
exceptional transparency of the sky at Dome C. A density filter was added on
each camera to avoid saturation by the daylight. The optical adjustment
procedure consists in aligning the focal box optical axis with the telescope
axis, which supposed to move the M1 mirror on three axis X,Y, and Z. Then, the
alignment of the equatorial mount was checked to ensure a perfect guiding on
the field for several hours. We cannot reach an alignment better than a few
arc minutes due to the small number of stars that can be observed during this
period of continuous daylight, and of the absence of a pointing model taking
into account the flexion of the telescope. This set-up is generally good
enough to ensure the operations, as the ASTEP+ field is about 40 arc minutes
wide, and a field recognition is done for each new observation. Real-time
guiding is also ensured from the image of the blue camera. However, for fields
close to the Celestial Pole, the uncertainty of the mount set-up results in a
noticeable field rotation, reducing the photometric quality of stars at the
edges of the image. In the future, we expect to have a new more reliable
mount, and to use an automatic set-up procedure to precisely align the mount
to within a fraction of arc minutes.

###  3.2 Turbulence measurement

To prevent frost from forming, the telescope primary and secondary mirrors are
heated from their back surface, as well parts of the camera focal box. This is
necessary although it is well known that heat sources in optical beam are
harmful to image quality. We ran a series of test in daytime to try to
quantify the effect of this heating on the turbulence inside the tube and the
dome, using a optical turbulence sensor, AIRFLOW [16]. This instrument
measures optical path differences along parallel beams within a sensing cell
by interferometric means, and by plotting the phase variance as a function of
beam separation, the phase structure function can be estimated. A value of
$C_{n}^{2}$ can be extracted if the phase structure function follows the
Kolmogorov structure, $D_{\phi}=(D/r_{0})^{5/3}$.

There is a small air gap at the bottom of the tube, allowing air to flow
around the primary mirror; without any active heating (but thermal driving
from sunlight) we found that the air inside the tube was very calm but some
mixing close due to this gap increased the turbulence around the primary
mirror (Figure 7). This illustrates that daytime tests are of limited
operational value, although they can be useful to provide guidelines in this
extreme environment.

---
Figure 7: Measurement of $C_{n}^{2}$ along the telescope tube in daytime. The
primary mirror is not heated but the mixing of air at different temperatures
around the air gap at the base of the tube increases the local turbulence.

We ran turbulence measurements during mirror heating with the dome both closed
and open (Figure 8). When the dome is closed, the environment around the
telescope is very stable and we are very sensitive to minute heating. However,
when the dome is open, which is much more representative of operational
conditions, even low wind clears out some heat and we need to increase the
temperature difference between the primary mirror and the ambient air to be
able to measure the turbulence. Unfortunately, when the dome is open, sunlight
differentially heats parts of the dome (we tried to prevent direct sunlight
hitting the telescope during those tests), contributing to local turbulence.

---
Figure 8: $C_{n}^{2}$ as a function of the temperature difference between the
primary mirror and the ambient air, measured just above the primary mirror
(left). When the dome is closed (middle), there is exquisite sensitivity to
$\Delta$T, while when the dome is closed (right), $C_{n}^{2}$ remains low for
small $\Delta$T, presumably due to flushing, and a transition to turbulence
occurs near $3^{\circ}C$.

Nonetheless we find that there is a regime (for $\Delta T<3^{\circ}C$) where
the air gap allows to flush the turbulence above the primary. The validity of
these experiments is limited in day time, and a ruggedized version of AIRFLOW
could be deployed in future observing seasons to provide quantitative control
of the primary mirror temperature.

###  3.3 Operations

The observation software was inherited from the previous version of the
instrument. Thanks to the modularity of the adopted software solution, the two
new 2K$\times$2K backside illuminated cameras (ANDOR iKon-L 936 and FLI Kepler
KL400) could be implemented rather easily. The main difference with the
previous version is that the blue channel KL400 camera is used both as a
guiding and a science detector. Short exposures (ranging from 0.1 to 5
seconds) are stacked to form $\sim$ 1 minute exposures. The particular sensor
operation of the KL400 allows digitizing non-destructively the same image with
two independent amplifier gain values using a 12 bits quantization (Low and
High gains). The idea behind this operation mode being to have a finer
sampling of lower intensities and a coarser sampling for higher intensities.
FLI uses a somewhat complex procedure to calibrate these two images and stitch
them together to form a full, 16 bits-equivalent image. In practice this
stitching procedure was found too complex to be implemented on ASTEP+, and the
non-linearity of each image resulted in non-satisfactory results for
photometric purposes. Instead, a custom solution relying on the
”linearization” of each frame using calibrated look-up-tables was envisaged
but not yet implemented. In the current version of the software, only the high
gain stacked images are used as science frames. The low gain images are used
for guiding on brighter nearby stars when possible since they are less prompt
to too high background levels situations, and the high gain images are used
when the single-frame exposure time needs to be very short (below 2 seconds).

###  3.4 Guiding

In the previous version of our instrument, a guiding camera was used solely
for guiding while ASTEP+ exploits the fast readout capability of the FLI KL400
camera to guide and save stacked science frames. Due to the mount inertia and
the mechanical backlash, the guiding frequency can hardly be higher than 1
Hertz. Based on tests performed at the beginning of the polar winter season we
set the guiding corrections frequency to 0.5 Hz. Typically when the target
star is brighter than V$\sim$11, the frame cadence is increased, but we keep
the mount guiding corrections to be below or equal to 0.5 Hz. Unfortunately,
with increasing nighttime duration and temperatures going down below -50∘C we
experienced random and too frequent communication losses between the
acquisition PC and the instrument. After thorough investigations, we
identified the USB3-to-Ethernet converter cable as the most probable cause of
these failures. The distance between the telescope and the laboratory where
the acquisition computer is hosted was about 60m, within the capability of the
USB-Ethernet converter. Disgracefully, the frost imposes special cables able
to resist to extreme temperature and the one we had was not sufficient for the
required specifications of the USB3 norm on this distance. To avoid frequent
interruptions in the acquisition, we decided to operate only the red camera,
and to update the guiding from the science images at a least frequent rate.
This proved to be efficient, and we did not noticed degradation of the
photometric accuracy.

An upgrade of this cable is foreseen for the next instrument update campaign.
In the meantime, we are investigating software solutions to mitigate the data
losses caused by frequent communication failures. Note that this sometimes
also affects the red Andor camera, but at a much lower rate (this camera uses
an USB2 protocol). In the future, the new direct-drive mount will allow more
frequent guiding updates which would also increase the precision.

###  3.5 Focusing

The focus of the images on the two detectors can be adjusted independently
thanks to a movable lens placed just before the camera. During the first
tests, the focus proved to be very sensitive to the external temperature, due
to the thermal dilatation of the telescope parts. The focus is determined
manually from the quality of the images and is adjusted regularly when the
temperature changes. An empirical temperature dependency law has been
determined. In the future, we foresee to set-up an automatic focus adjustment
procedure.

##  4 FIRST RESULTS

One of the important characterization of the new focal box consists in the
comparison between the theoretical transmission of its two photometric
channels with other photometric measurements, particularly from space. In
Figure 9, we present the response of the two-color channels together with the
estimated response of the GAIA filters. The transmissions of the blue and red
channels of the new focal box are almost identical to the GAIA blue and red
channels respectively, although both channels are slightly redder. The
transmission of the GAIA red filter is also much higher. Figure 10 shows the B
and R flux of the stars in the ASTEP field (here the WASP 131 field)
normalised to the brightest star of the field as compared to the GAIA B and R
magnitudes. It shows an almost unity slope of the flux to the magnitude, so
without biased estimation, and with a different vertical offset due to the
difference in transmission.

To be more precise in the color magnitude determination, figure 11 shows a
comparison between B-R measured on all stars of the field of WASP131 using
SExtractor software [17] to the color magnitude B-R of the same stars in the
GAIA catalogue.The B-R color response of the new focal box to stars of
different effective temperatures is also almost the same as that of GAIA, with
a different zero point. The magnitude difference between ASTEP+ B-R and GAIA
is estimated to 1.5 magnitude.

---
Figure 9: Transmission of the two channels of the new focal box calculated
from theoretical transmission curves as compared to the GAIA blue and red
channels.
---
Figure 10: Comparison of the color magnitude B (resp. R) measured on stars of
the field of WASP131 as compared to the value B (resp. R) of the same stars in
the GAIA catalogue.
---
Figure 11: B-R ASTEP+ magnitude computed on the same field and compared to the
B-R magnitudes of the stars from the GAIA catalogue.

After a period of tests and adjustment, normal operations started at the end
of March 2022. An usual observing sequence permits to follow the field around
a given star (usually a star known to exhibit transiting exoplanets) for a
period of several hours. These observations permit to confirm the parameters
of the transit (depth, duration, mean time) and to exclude false detection due
to background binaries. It also allows the determination of updated
ephemerides for future observations by space mission. When close to the winter
solstice, the observations could be almost continuous, allowing the
observation of very long transits.

Figure 12 shows an example of lightcurves in two colors obtained with the new
focal box. Here, we show the transit of WASP-19b that we already observed in
2013 with a single camera[18]. The transit has been recorded in two colors,
blue and red, with a similar precision. The dispersion of the measurement is
of 0.80 ppt (parts per thousand) and 1.19 ppt for an integration of 2 minutes
in the Blue and in the Red channels respectively. This has to be compared to
the previous dispersion of 1.7 ppt on the same star observed in 2021 with the
previous camera. It can be seen that the transit is slightly in advance with
respect to the known ephemerides, showing the need for follow-up with
instruments such as ASTEP+ to refine the ephemerides for future space missions
and to determine possible Transit Timing Variations (TTV) due to the presence
of other planets in the system.

---
Figure 12: Example of a two color photometric transit of exoplanet WASP-19b
recorded at Dome C with ASTEP+ on 2022-04-11. The red points correspond to the
data obtained in the ASTEP+ red channel. The blue points correspond to the
data obtained in the ASTEP+ blue channel. Note that the transit is achromatic
(as expected), and it occurs $\sim 7$ mns earlier than expected, a consequence
of the accumulation of errors on the ephemerides.

##  5 CONCLUSION

A new two-color focal box was successfully installed on the ASTEP telescope at
Concordia station for the follow-up of transiting exoplanets. Despite a
failure in the communication with one of the camera, the installation was
realized without major issues. The new focal box is heavier than the previous
one, so we could fear that the guiding could be problematic, as the mount had
shown problems in the past. To the contrary, the telescope behaviour and the
first results show no problems with the guiding. The quality of the data, even
with one camera, is significantly improved with respect to previous
observations. With two cameras simultaneously, the determination of transits
would be more precise

The cause of the communication issue with the blue camera has been identified
and will be solved by next year, allowing a complete use of the focal box. The
comparison of the color magnitudes of ASTEP+ with GAIA shows an excellent
correlation and allows to determine absolute magnitudes from ASTEP+ data.

In the future, we will also change the telescope mount for a more powerful
one, without gears, allowing faster pointing and guiding which should increase
again the performance and allow other type of observing programs. This new
mount should be installed in 2024.

## ACKNOWLEDGMENTS

ASTEP and ASTEP+ have benefited from the support of the French and Italian
polar agencies IPEV and PNRA, and by the Université Côte d’Azur through Idex
UCAJEDI (ANR-15-IDEX-01). We acknowledge support from the European Space
Agency (ESA) through the Science Faculty of the European Space Research and
Technology Centre (ESTEC). The realisation and installation of the new ASTEP+
focal box was possible thanks to the support of ESA ESTEC and in part funded
by the European Union’s Horizon 2020 research and innovation programme (grants
agreements n∘ 803193/BEBOP), from the Science and Technology Facilities
Council (STFC; grant n∘ ST/S00193X/1), from INSU, and from the laboratoire
Lagrange (CNRS UMR 7293).

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# Lagrangian densities of some $3$-uniform hypergraphs

Zilong Yan School of Mathematics, Hunan University, Changsha 410082, P.R.
China. Email<EMAIL_ADDRESS>Yuejian Peng  Corresponding author. School
of Mathematics, Hunan University, Changsha, 410082, P.R. China. Email:
<EMAIL_ADDRESS>Supported in part by National Natural Science Foundation of
China (No. 11931002 and No. 11671124).

###### Abstract

The Lagrangian density of an $r$-uniform hypergraph $H$ is $r!$ multiplying
the supremum of the Lagrangians of all $H$-free $r$-uniform hypergraphs. For
an $r$-uniform graph $H$ with $t$ vertices, it is clear that
$\pi_{\lambda}(H)\geq r!\lambda{(K_{t-1}^{r})}$. We say that an $r$-uniform
hypergraph $H$ with $t$ vertices is $\lambda$-perfect if
$\pi_{\lambda}(H)=r!\lambda{(K_{t-1}^{r})}$. A theorem of Motzkin and Straus
implies that all $2$-uniform graphs are $\lambda$-perfect. It is interesting
to understand what kind of hypergraphs are $\lambda$-perfect. The property
‘$\lambda$-perfect’ is monotone in the sense that an $r$-graph obtained by
removing an edge from a $\lambda$-perfect $r$-graph (keep the same vertex set)
is $\lambda$-perfect. It’s interesting to understand the relation between the
number of edges in a hypergraph and the ‘$\lambda$-perfect’ property. We
propose that the number of edges in a hypergraph no more than the number of
edges in a linear hyperpath would guarantee the ‘$\lambda$-perfect’ property.
We show some partial result to support this conjecture. We also give some
partial result to support the conjecture that the disjoint union of two
$\lambda$-perfect $r$-uniform hypergraph is $\lambda$-perfect. We show that
the disjoint union of a $\lambda$-perfect $3$-graph and
$S_{2,t}=\\{123,124,125,126,...,12(t+2)\\}$ is perfect. This result implies
the earlier result of Heftz and Keevash, Jiang, Peng and Wu, and several other
earlier results.

Let $\Pi_{r}=\\{\pi({\mathcal{F}}):{\mathcal{F}}{\rm\ is\ a\ family\ of\
}r-{\rm uniform\ graphs}.\\}$, where $\pi({\mathcal{F}})$ is the Turán density
of ${\mathcal{F}}$. Erdős-Stone-Simonovits showed that $\Pi_{2}=\\{0,{1\over
2},{2\over 3},{3\over 4},\ldots\\}$. However, very few are known for $\Pi_{r}$
when $r\geq 3$. Sidorenko showed that the Lagrangian density of an $r$-uniform
graph is the same as the Turán density of its extension. In particular, these
two densities of $H$ equal if every pair of vertices of $H$ is contained in an
edge. Sidorenko gave the Lagrangian densities of infinitely many hypergraphs
and this led to first know infinitely many numbers in $\Pi_{r}$ for $r\geq 3$.
Determining the Turán density of $K_{4}^{3-}$ (a well-known conjecture of
Frankl and Füredi) is one of the major open problems in hypergraph Turán
problems. The result of Sidorenko implies that the Lagrangian density and the
Turán density of $K_{4}^{3-}$ equal. It would be very interesting to obtain
the Turán density of $K_{4}^{3-}$ by determining the Lagrangian density of
$K_{4}^{3-}$. As a first attempt, we determine the Lagrangian density of
$K_{4}^{3-}\sqcup e$, where $e$ is a single edge. We also find an irrational
Lagrangian density. The advantage of transferring to the Lagrangian density is
that we can assume that $G$ is ‘dense’ (so covering pairs) when considering
the maximum Lagrangian of an $H$-free $r$-graph $G$. This assumption makes the
structural analysis ‘nicer’ in some cases, so we hope that this method helps
us to better understand the set $\Pi_{r}$.

Key Words: Hypergraph Lagrangian

##  1 Introduction

###  1.1  Notations and definitions

For a set $V$ and a positive integer $r$, let $V^{r}$ denote the family of all
$r$-subsets of $V$. An $r$-uniform graph or $r$-graph $G$ consists of a set
$V(G)$ of vertices and a set $E(G)\subseteq V(G)^{r}$ of edges. Let $|G|$
denote the number of edges of $G$. An edge $e=\\{a_{1},a_{2},\ldots,a_{r}\\}$
will be simply denoted by $a_{1}a_{2}\ldots a_{r}$. An $r$-graph $H$ is a
subgraph of an $r$-graph $G$, denoted by $H\subseteq G$, if $V(H)\subseteq
V(G)$ and $E(H)\subseteq E(G)$. A subgraph of $G$ induced by
$V^{\prime}\subseteq V$, denoted as $G[V^{\prime}]$, is the $r$-graph with
vertex set $V^{\prime}$ and edge set $E^{\prime}=\\{e\in E(G):e\subseteq
V^{\prime}\\}$. For $S\subseteq V(G)$, let $G-S$ denote the subgraph of $G$
induced by $V(G)\setminus S$. Let $K^{r}_{t}$ denote the complete $r$-graph on
$t$ vertices. Let $K^{r-}_{t}$ be obtained by removing one edge form the
complete $r$-graph on $t$ vertices. For a positive integer $n$, let $[n]$
denote $\\{1,2,3,\ldots,n\\}$.

Given an $r$-graph $F$, an $r$-graph $G$ is called $F$-free if it does not
contain an isomorphic copy of $F$. For a fixed positive integer $n$ and an
$r$-graph $F$, the Turán number of $F$, denoted by $ex(n,F)$, is the maximum
number of edges in an $F$-free $r$-graph on $n$ vertices. An averaging
argument of Katona, Nemetz and Simonovits [14] shows that the sequence
${ex(n,F)\over{n\choose r}}$ is non-increasing. Hence
$\lim_{n\rightarrow\infty}{ex(n,F)\over{n\choose r}}$ exists. The Turán
density of $F$ is defined as

$\pi(F)=\lim_{n\rightarrow\infty}{ex(n,F)\over{n\choose r}}.$

Denote

$\Pi_{r}=\\{\pi({\mathcal{F}}):{\mathcal{F}}{\rm\ is\ a\ family\ of\ }r-{\rm
uniform\ graphs}.\\}.$

For 2-graphs, Erdős-Stone-Simonovits determined the Turán numbers of all non-
bipartite graphs asymptotically. Their Result implies that
$\Pi_{2}=\\{0,{1\over 2},{2\over 3},{3\over 4},\ldots\\}$. Very few results
are known for $r\geq 3$ and a survey on this topic can be found in Keevash’s
survey paper [15]. The hypergraph Lagrangian method has been helpful in
hypergraph extremal problems.

###### Definition 1.1

Let $G$ be an $r$-graph on $[n]$ and let
$\vec{x}=(x_{1},\ldots,x_{n})\in[0,\infty)^{n}$. Define the Lagrangian
function

$\lambda(G,\vec{x})=\sum_{e\in E(G)}\prod\limits_{i\in e}x_{i}.$

The Lagrangian of $G$, denoted by $\lambda(G)$, is defined as

$\lambda(G)=\max\\{\lambda(G,\vec{x}):\vec{x}\in\Delta\\},$

where

$\Delta=\\{\vec{x}=(x_{1},x_{2},\ldots,x_{n})\in[0,1])^{n}:x_{1}+x_{2}+\dots+x_{n}=1\\}.$

The value $x_{i}$ is called the weight of the vertex $i$ and a vector
$\vec{x}\in{\Delta}$ is called a feasible weight vector on $G$. A feasible
weight vector $\vec{y}\in{\Delta}$ is called an optimum weight vector for $G$
if $\lambda(G,\vec{y})=\lambda(G)$.

In [17], Motzkin and Straus established a connection between the Lagrangian of
a $2$-graph and it’s maximum complete subgraphs.

###### Theorem 1.2

([17]) If $G$ is a $2$-graph in which a maximum complete subgraph has $t$
vertices, then $\lambda(G)=\lambda(K_{t}^{2})={1\over 2}(1-{1\over t})$.

They also applied this connection to give another proof of the theorem of
Turán on the Turán density of complete graphs. Since then the Lagrangian
method has been a useful tool in hypergraph extremal problems. Earlier
applications include that Frankl and Rödl [9] applied it in disproving the
long standing jumping constant conjecture of Erdős. Sidorenko [24] applied
Lagrangians of hypergraphs to first find infinitely many Turán densities of
hypergraphs. More recent developments of this method were obtained in [20, 2,
10, 18, 4, 19, 12, 13, 11, 30]. In addition to its applications in extremal
problems, it is interesting in its own right to determine the maximum
Lagrangian of $r$-graphs with certain properties as remarked by Hefetz and
Keevash [10]. For example, an interesting conjecture of Frankl-Füredi [8]
states that the the maximum Lagrangian among all $r$-graphs with $m$ edges
achieves on the $r$-graph whose edges are the first $r$-tuples in colex order.
Talbot [25] made a first breakthrough in confirming this conjecture for some
cases. Subsequent progress were made in [27, 16, 26]. Recently, Gruslys-
Letzter-Morrison [GLM2018] confirmed this conjecture for $r=3$ and
sufficiently large $m$, and showed that the conjecture is not always true for
$r\geq 4$. We focus on the Lagrangian density of an $r$-graph in this paper.

The Lagrangian density  $\pi_{\lambda}(F)$ of an $r$-graph $F$ is defined to
be

$\pi_{\lambda}(F)=\sup\\{r!\lambda(G):G\>\>{\rm is}\>\>F{\text{-}}{\rm
free}\\}.$

A pair of vertices $\\{i,j\\}$ is covered in a hypergraph $F$ if there exists
an edge $e$ in $F$ such that $\\{i,j\\}\subseteq e$. We say that $F$ covers
pairs if every pair of vertices in $F$ is covered. Let $r\geq 3$ and $F$ be an
$r$-graph. The extension of $F$, denoted by $H^{F}$ is obtained as follows:
For each pair of vertices $v_{i}$ and $v_{j}$ not covered in $F$, we add a set
$B_{ij}$ of $r-2$ new vertices and the edge $\\{v_{i},v_{j}\\}\cup B_{ij}$,
where the $B_{ij}$’s are pairwise disjoint over all such pairs $\\{i,j\\}$.

The Lagrangian density is closely related to the Turán density. The following
proposition is implied by Theorem $2.6$ in [23] (see Proposition $5.6$ in [4]
and Corollary $1.8$ in [24] for the explicit statement).

###### Proposition 1.3

([23, 4, 24]) Let $F$ be an $r$-graph. Then
$(i)$ $\pi(F)\leq\pi_{\lambda}(F);$
$(ii)$ $\pi(H^{F})=\pi_{\lambda}(F).$ In particular, if $F$ covers pairs, then
$\pi(F)=\pi_{\lambda}(F).$

The Lagrangian density of the enlargement of a tree satisfying Erdős-Sos’s
conjecture is determined by Sidorenko [24] and Brandt-Irwin-Jiang [4]. This
first gave the Turán densities of infinitely many hypergraphs. Pikhurko [20]
determined the Lagrangian density of a $4$-uniform tight path of length 2 and
this led to confirm the conjecture of Frankl-Füredi on the Turán number of its
extension, the $r$-uniform genearlized triangle for the case $r=4$. Norin and
Yepremyan [19] determined for $r=5$ or $6$ by extending the earlier result of
Frankl-Füredi in [8]. Jenssen [12] determined the Lagrangian density of a path
of length 2 formed by two edges intersecting at $r-2$ vertices for
$r=3,4,5,6,7$. Hefetz and Keevash [10] determined the Lagrangian density of a
$3$-uniform matching of size 2. Jiang-Peng-Wu [13] obtained for any
$3$-uniform matching. The case for an $r$-uniform matching of size 2 was given
in [3] (independently, in [28] for $r=4$). In [29, 11, 5, 30], the authors
obtained the Lagrangian densities of a $3$-uniform linear path of length $3$
or $4$, the Lagrangian densities of $\\{123,234,456\\}$, {123,345,561} and
{123,124,345},the disjoint union of a $3$-uniform linear path of length $2$ or
$3$ and a $3$-uniform matching, and the disjoint union of a $3$-uniform tight
path of length $2$ and a $3$-uniform matching. These were all the previously
known results on Lagrangian densities. For $3$-uniform graphs spanned by 3
edges, there is one remaining unsolved case: $K_{4}^{3-}=\\{123,124,134\\}$.
By Proposition 1.3, the Lagrangian density and the Turán density of
$K_{4}^{3-}$ equal. It would be very interesting to obtain the Turán density
of $K_{4}^{3-}$ (a well-known conjecture of Frankl and Füredi) by determining
the Lagrangian density of $K_{4}^{3-}$.

###  1.2  Main results and remarks

For an $r$-graph $H$ on $t$ vertices, it is clear that $\pi_{\lambda}(H)\geq
r!\lambda{(K_{t-1}^{r})}$.

###### Definition 1.4

An $r$-graph $H$ on $t$ vertices is $\lambda$-perfect if
$\pi_{\lambda}(H)=r!\lambda{(K_{t-1}^{r})}$.

Theorem 1.2 implies that all $2$-graphs are $\lambda$-perfect. It is
interesting to explore what kind of hypergraphs are $\lambda$-perfect. An
$r$-uniform hypergraph is linear if any two edges have at most 1 vertex in
common. Let $G\bigsqcup H$ denote the disjoint union of $G$ and $H$. The
following conjectures were proposed in [30].

###### Conjecture 1.5

[30] For $r\geq 3$, there exists $n$ such that a linear $r$-graph with at
least $n$ vertices is $\lambda$-perfect.

###### Conjecture 1.6

[30] For $r\geq 3$, there exists $n$ such that if $G$ and $H$ are
$\lambda$-perfect $r$-graphs with at least $n$ vertices, then the disjoint
union of $G$ and $H$ is $\lambda$-perfect.

Let $S_{2,t}$ denote the 3-graph with vertex set
$\\{v_{1},v_{2},u_{1},u_{2},...,u_{t}\\}$ and edge set
$\\{v_{1}v_{2}u_{1},v_{1}v_{2}u_{2},...,v_{1}v_{2}u_{t}\\}$. A result of
Sidorenko in [24] implies that $S_{2,t}$ is $\lambda$-perfect. In [30], we
proved that $S_{2,t}\sqcup H$ is $\lambda$-perfect if $H$ is $\lambda$-perfect
and $t\geq 3$. Our first result in this paper removes the condition that
$t\geq 3$.

###### Theorem 1.7

If H is $\lambda$-perfect, then $H\sqcup S_{2,t}$ is $\lambda$-perfect for any
$t\geq 1$.

Taking $H$ to be the 3-graph spanned by one edge or the 3-uniform linear path
of length 2 or $3$ , and $t=1$, and applying the above result, we can obtain
the results in [10, 13, 11]. The condition that H is $\lambda$-perfect in the
above theorem is not necessary.

Let $F_{5}$ denote the 3-graph with vertex set
$\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}$ and edge set
$\\{v_{1}v_{2}v_{3},v_{1}v_{2}v_{4},v_{3}v_{4}v_{5}\\}$. In [30], we proved
that $\pi_{\lambda}(F_{5})=\frac{4}{9}$ and this implies that $F_{5}$ is not
$\lambda$-perfect. We show that $F_{5}\sqcup S_{2,t}$ is $\lambda$-perfect.
Let $H\sqcup\\{e\\}$ denote the disjoint union of $H$ and a single edge
throughout the paper.

###### Theorem 1.8

$F_{5}\sqcup\\{e\\}$ is $\lambda$-perfect.

###### Theorem 1.9

$F_{5}\sqcup S_{2,t}$ is $\lambda$-perfect for $t\geq 2$.

Determining the Turán density of $K_{4}^{3-}$ (a well-known conjecture of
Frankl and Füredi) is one of the major open problems in hypergraph Turán
problems. By Proposition 1.3, the Lagrangian density and the Turán density of
$K_{4}^{3-}$ equal. It would be very interesting to obtain the Turán density
of $K_{4}^{3-}$ by determining the Lagrangian density of $K_{4}^{3-}$. As a
first attempt, we determine the Lagrangian density of $K_{4}^{3-}\sqcup e$. We
show the following result.

###### Theorem 1.10

$K_{4}^{3-}\sqcup S_{2,t}$ is $\lambda$-perfect for any $t\geq 1$.

We also discuss $H\sqcup S_{2,t}$ for any $3-$graph $H$.

###### Theorem 1.11

Let H be a $3$-graph with $s$ vertices. Then $H\sqcup S_{2,t}$ is
$\lambda$-perfect if $t\geq\frac{3}{2}s^{2}-\frac{11}{2}s+4$.

Note that the property ‘$\lambda$-perfect’ is monotone in the sense that an
$r$-graph obtained by removing an edge from a $\lambda$-perfect $r$-graph
(keep the same vertex set) is $\lambda$-perfect. It’s interesting to
understand the relation between the number of edges in a hypergraph and the
‘$\lambda$-perfect’ property. We propose that the number of edges in a
hypergraph is no more than the number of edges in a linear hyperpath would
guarantee the ‘$\lambda$-perfect’ property.

###### Conjecture 1.12

For $r\geq 3$, there exists $m_{0}$ such that for an $r$-graph $G$ with $m\geq
m_{0}$ edges, if the number of vertices in $G$ is at least $m(r-1)+1$, then
$G$ is $\lambda$-perfect.

Theorems 1.7 to 1.11 give some evidence to this conjecture. We remark that a
hypergraph is not $\lambda$-perfect if it has many edges.

###### Remark 1.13

Let H be a 3-graph on t vertices with at least
$\binom{t-1}{3}+\binom{t-2}{2}+2$ edges. Then H is not $\lambda$-perfect.

Proof. Let G be a 3-graph with vertex set $[t]$ and edge set
$\\{ijk|i,j,k\in[t-1]\\}\cup\\{ijt|i,j\in[t-2]\\}\cup\\{1(t-1)t\\}$. Note that
$G$ has $\binom{t-1}{3}+\binom{t-2}{2}+1$ edges , so G is H-free. Let
$x_{i}=\frac{1}{t-1}$ for $1\leq i\leq t-2$, $x_{j}=\frac{1}{2t-2}$ for
$t-1\leq j\leq t$, and $\vec{x}$ be a feasible vector such that vertex i has
weight $x_{i}$. Then
$\lambda(G,\vec{x})=\frac{1}{(t-1)^{3}}\bigg{[}\binom{t-1}{3}+\frac{1}{4}\bigg{]}>\frac{1}{(t-1)^{3}}\binom{t-1}{3}=\lambda(K_{t-1}^{3})$.
Therefore H is not $\lambda$-perfect. $\Box$

It might be interesting to understand whether the property ‘covering pairs’
plays some role for the ‘$\lambda$-perfect’ property. Proposition 1.3 says
that the property ‘covering pairs’ is a sufficient condition for the
Lagrangian density to be the same as the Turán density. Is it a necessary
condition?

###### Question 1

Whether an $r$-graph $G$ covering pairs implies that $G$ is not
$\lambda$-perfect?

###### Question 2

Whether an $r$-graph $G$ covering pairs implies that
$\pi(G)\neq\pi_{\lambda}(G)$ ?

In contrast to the case $r=2$, we know very few about the set $\Pi_{r}$ for
$r\geq 3$. By Proposition 1.3, if we get a Lagrangian density, then we can get
a corresponding Turán density. Sidorenko [24] first determined infinite many
numbers in $\Pi_{r}$ for $r\geq 3$ by showing the Lagrangian densities of an
infinite family of hypergraphs. Researching on Lagrangian densities on
hypergraphs might help us understand the set $\Pi_{r}$ better.

In 1999, Chung and Graham [6] proposed the conjecture that every element in
$\Pi_{r}$ is a rational number. This conjecture was recently disproved by
Baber and Talbot [2], and Pikhurko [21] independently by showing that a family
of several r-graphs has irrational Turán density. Baber and Talbot [2] asked
whether there exists an r-graph F such that $\pi(F)$ is irrational. As
mentioned earlier, the Lagrangian density and the Turán density of $K_{4}^{3}$
equal. Determining the Lagrangian density of $K_{4}^{3}$ would lead to the
solution of the long-standing conjecture of Turán on the Turán density of
$K_{4}^{3}$. As a first attempt, we try to determine the Lagrangian density of
$K_{4}^{3}\cup e$. We conjecture that the Lagrangian density of $K_{4}^{3}\cup
e$ achieves on the $3$-graph $S_{2}(n)$ with vertex set $[n]$ and edge set
$\\{12i|i\in[n]\setminus\\{1,2\\}\\}\cup\\{ijk|i\in[2],j,k\in[n]\setminus\\{1,2\\}\\}$
as $n\rightarrow\infty$. If this is true, then the Lagrangian density of
$K_{4}^{3}\cup e$ is $\frac{\sqrt{3}}{3}$, and this would answer the question
of Baber and Talbot. But due to some difficulty, we can only show the
following result and add one more irrational number to $\Pi_{3}$.

###### Theorem 1.14

Let $\mathscr{P}=\\{P_{1},P_{2},P_{3},P_{4}\\}$ where
$P_{1}=\\{123,124,134,234,567\\}$, $P_{2}=\\{123,124,134,234,\\\
561,562,783\\}$, $P_{3}=\\{123,124,134,234,561,562,734\\}$ and
$P_{4}=\\{123,124,134,234,561,562,357\\}$. Then
$\pi_{\lambda}(\mathscr{P})=\frac{\sqrt{3}}{3}$.

As stated in Proposition 1.3, the Lagrangian density and the Turán density are
the same for a large class of $r$-graphs.The advantage of transferring to the
Lagrangian density is that we can assume that $G$ is dense (so covering pairs)
when considering the maximum Lagrangian of an $H$-free $r$-graph $G$. This
assumption makes the structural analysis ‘nicer’ in some cases, so we hope
that this method helps us to better understand the set $\Pi_{r}$.

We give some properties of Lagrangians of $r$-graphs in the next section. The
proofs of Theorems 1.7 to 1.11 will be given in Section 3. The proof of
Theorem 1.14 will be given in Section 4.

##  2 Preliminaries

The following fact follows immediately from the definition of the Lagrangian.

###### Fact 2.1

Let $G_{1}$, $G_{2}$ be $r$-graphs and $G_{1}\subseteq G_{2}$. Then
$\lambda(G_{1})\leq\lambda(G_{2}).$

###### Fact 2.2

([9]) Let $G$ be an $r$-graph on $[n]$. Let
$\vec{x}=(x_{1},x_{2},\dots,x_{n})$ be an optimum weight vector on $G$. Then

$\frac{\partial\lambda(G,\vec{x})}{\partial x_{i}}=r\lambda(G)$

for every $i\in[n]$ satisfying $x_{i}>0$.

Given an $r$-graph $G$, and $i,j\in V(G),$ define

$L_{G}(j\setminus i)=\\{e:i\notin e,e\cup\\{j\\}\in
E(G)\>and\>e\cup\\{i\\}\notin E(G)\\}.$

###### Fact 2.3

Let $G$ be an $r$-graph on $[n]$. Let $\vec{x}=(x_{1},x_{2},\dots,x_{n})$ be a
feasible weight vector on $G$. Let $i,j\in[n]$, $i\neq j$ satisfying
$L_{G}(i\setminus j)=L_{G}(j\setminus i)=\emptyset$. Let
$\vec{y}=(y_{1},y_{2},\dots,y_{n})$ be defined by letting $y_{\ell}=x_{\ell}$
for every $\ell\in[n]\setminus\\{i,j\\}$ and $y_{i}=y_{j}={1\over
2}(x_{i}+x_{j})$. Then $\lambda(G,\vec{y})\geq\lambda(G,\vec{x})$.
Furthermore, if the pair $\\{i,j\\}$ is contained in an edge of $G$, $x_{i}>0$
for each $1\leq i\leq n$, and $\lambda(G,\vec{y})=\lambda(G,\vec{x})$, then
$x_{i}=x_{j}$.

Proof. Since $L_{G}(i\setminus j)=L_{G}(j\setminus i)=\emptyset$, then

$\lambda(G,\vec{y})-\lambda(G,\vec{x})=\sum_{\\{i,j\\}\subseteq e\in
G}\left({(x_{i}+x_{j})^{2}\over 4}-x_{i}x_{j}\right)\prod\limits_{k\in
e\setminus\\{i,j\\}}x_{k}\geq 0.$

If the pair $\\{i,j\\}$ is contained in an edge of $G$ and $x_{i}>0$ for each
$1\leq i\leq n$, then the equality holds only if $x_{i}=x_{j}$. $\Box$

An $r$-graph $G$ is dense if $\lambda(G^{\prime})<\lambda(G)$ for every proper
subgraph $G^{\prime}$ of $G$.

###### Fact 2.4

([9]) Let $G=(V,E)$ be a dense $r$-graph. Then $G$ covers pairs.

Note that the converse of Fact 2.4 is not true. For example, the Fano plane
covers pairs but it is not dense. Indeed, many counterexamples exist by
Theorem 2.1 in the paper of Talbot [25]. While considering the Lagrangian
density of an $r$-graph $F$, we can always reduce to consider dense $F$-free
$r$-graphs.

###### Remark 2.5

Let $F$, $G$ be $r$-graphs and $G$ be $F$-free. Then (a) there exists a dense
subgraph $G^{\prime}$ of $G$ such that $\lambda{(G^{\prime})}=\lambda{(G)}$
and $G^{\prime}$ is $F$-free. (b) To show $\pi_{\lambda}(F)\leq a$, it’s
sufficient to show that $\lambda(G)\leq\frac{a}{r!}$ for any dense F-free
r-graph. (c) To show that a t-vertex r-graph is $\lambda$-perfect, it’s
sufficient to show that $\lambda(G)\leq\lambda(K_{t-1}^{r})$ for any dense
F-free r-graph.

Proof. (a). Let $G$ be an $r$-graph on $n$ vertices. If $G$ is dense, then we
are fine. If not, then we can find $G^{\prime}\subset G$ such that
$\lambda{(G^{\prime})}=\lambda{(G)}$ and $|V(G^{\prime})|<|V(G)|.$ If
$G^{\prime}$ is dense, then we stop. Otherwise, we continue this process until
we find a dense subgraph. This process terminates since the number of vertices
is reduced by at least one in each step.

(b) and (c) follows immediately from (a). $\Box$

##  3 The disjoint union of a 3-graph and $S_{2,t}$

We will prove Theorems 1.7 to 1.11 in this section.

###  3.1 Preliminaries

For a 3-graph G and $v\in V(G)$, we define the link graph of v as
$G_{v}=\\{ab|vab\in E(G)\\}$. Let $\omega(G)$ denote the order of a maximum
clique of G.

###### Claim 3.1

Let G be a 3-graph with $\lambda(G)>\lambda(K_{k+1}^{3})$ and let $\vec{x}$ be
an optimal weight vector. Then for any $v\in V(G)$, its weight $x_{v}$
satisfies that $x_{v}<1-\frac{\sqrt{k(k-1)}}{k+1}$.

_Proof._ Since the link graph $G_{v}$ of v is a graph, by Fact 2.2 and the
theorem of Motzkin and Straus (Theorem 1.2), we have

$3\lambda(K_{k+1}^{3})<3\lambda(G)=\frac{\partial\lambda(G)}{\partial
x_{v}}\leq\frac{1}{2}(1-x_{v})^{2}.$

So

$x_{v}<1-\frac{\sqrt{k(k-1)}}{k+1}.$

$\Box$

###### Claim 3.2

Let G be a 3-graph with $\lambda(G)>\lambda(K_{k+1}^{3})$ and let $\vec{x}$ be
an optimal weight vector. Then for $v\in V(G)$ with $\omega(G_{v})\leq k$, its
weight $x_{v}$ satisfies that $x_{v}<\frac{1}{k+1}$.

_Proof._ Since the link graph $G_{v}$ of v is a graph, by Fact 2.2 and the
theorem of Motzkin and Straus (Theorem 1.2), we have

$3\lambda(K_{k+1}^{3})<3\lambda(G)=\frac{\partial\lambda(G)}{\partial
x_{v}}\leq\frac{1}{2}\times(1-\frac{1}{k})(1-x_{v})^{2}.$

So

$x_{v}<\frac{1}{k+1}.$

$\Box$

###### Claim 3.3

Let v be a vertex in a 3-graph G and $x_{v}$ be the weight of v in an optimal
weight vector $\vec{x}$ of G. If $G-\\{v\\}$ is H-free, then
$\lambda(G)\leq\frac{\pi_{\lambda}(H)(1-x_{v})^{3}}{6(1-3x_{v})}$.

_Proof._ Since $G-\\{v\\}$ is H-free, then
$\lambda(G-\\{v\\},\vec{x})\leq\frac{\pi_{\lambda}(H)}{6}(1-x_{v})^{3}$.
Therefore

$\lambda(G)\leq\frac{\pi_{\lambda}(H)}{6}(1-x_{v})^{3}+x_{v}\frac{\partial{\lambda(G)}}{\partial
x_{v}}.$

By Fact 2.2, we have

$\lambda(G)\leq\frac{\pi_{\lambda}(H)}{6}(1-x_{v})^{3}+3x_{v}\lambda(G).$

Then $\lambda(G)\leq\frac{\pi_{\lambda}(H)(1-x_{v})^{3}}{6(1-3x_{v})}$. $\Box$

###### Remark 3.4

$f(x)=\frac{(1-x)^{3}}{1-3x}$ is increasing in $(0,\frac{1}{3}).$

_Proof._ Since $f^{\prime}(x)=\frac{6x(1-x)^{2}}{(1-3x)^{2}}$, then
$f^{\prime}(x)>0$ in $(0,\frac{1}{3}).$ So $f(x)=\frac{(1-x)^{3}}{1-3x}$ is
increasing in $(0,\frac{1}{3}).$ $\Box$

###### Claim 3.5

Let a 3-graph G be $H\cup S_{2,t}$-free, where H is a 3-graph with s vertices.
Let
$S_{2,s+t}=\\{v_{1}v_{2}b_{1},v_{1}v_{2}b_{2},...,v_{1}v_{2}b_{s+t}\\}\subseteq
G$. Then $G-\\{v_{1},v_{2}\\}$ is H-free.

_Proof._ Suppose that $H\subseteq G-\\{v_{1},v_{2}\\}$. Since $|V(H)|=s$, then
$|\\{b_{1},b_{2},...,b_{s+t}\\}\cap V(H)|\leq s$, and
$|\\{b_{1},...,b_{s+t}\\}\setminus V(H)|\geq t$. So the induced subgraph of G
by $\\{v_{1},v_{2},b_{1},...,b_{s+t}\\}\setminus V(H)$ contains an $S_{2,t}$,
and G contains $H\cup S_{2,t}.$ $\Box$

###### Claim 3.6

Let a 3-graph G be $H\cup S_{2,t}$-free, where H is a 3-graph with s vertices.
Let $v\in V(H)$. If $H\subseteq G-\\{v\\}$, then $\omega(G_{v})\leq s+t$.

_Proof._ Since $G-\\{v\\}$ contains H. If $\omega(G_{v})\geq s+t+1,$ and
assume that a maximum clique of $G_{v}$ has vertex set $U_{1}$. Since
$|U_{1}\cap V(H)|\leq s,$ then $|U_{1}\setminus V(H)|\geq t+1$ and the induced
subgraph of G by $U_{1}\cup\\{v_{1}\\}\setminus V(H)$ contains an $S_{2,t}$.
So $H\cup S_{2,t}\subseteq G$, a contradiction. $\Box$

###### Claim 3.7

Let a 3-graph G be $H\cup S_{2,t}$-free, where H is a 3-graph with s vertices.
If $H\subseteq G-\\{v_{1}\\}$ and $H\nsubseteq G-\\{v_{1},v_{2}\\}$, then
$\omega((G-\\{v_{2}\\})_{v_{1}})\leq s+t-1.$

_Proof._ Assume that $\omega((G-\\{v_{2}\\})_{v_{1}})\geq s+t$ and a maximum
clique of $(G-\\{v_{2}\\})_{v_{1}}$ has vertex set $U_{2}$. Since $H\subseteq
G-\\{v_{1}\\}$ and $H\nsubseteq G-\\{v_{1},v_{2}\\}$, then $v_{2}\in V(H)$.
Therefore $|U_{2}\cap V(H)|\leq s-1.$ So $|U_{2}\setminus V(H)|\geq t+1$ and
the induced subgraph of G by $U_{2}\cup\\{v_{1}\\}\setminus V(H)$ contains an
$S_{2,t}$. Therefore $H\cup S_{2,t}\subseteq G$. $\Box$

###### Theorem 3.8

([1][22]) $\pi(K_{4}^{3-})\leq 0.2871$, $\pi(K_{4}^{3})\leq 0.5617.$

###  3.2 The disjoint union of a $\lambda$-perfect 3-graph and $S_{2,t}$

We show that the disjoint union of a $\lambda$-perfect 3-graph and $S_{2,t}$
is $\lambda$-perfect.

_Proof of Theorem 1.7._ Assume that H is $\lambda$-perfect on $s\geq 3$
vertices. Note that $H\cup S_{2,t}$ has s+t+2 vertices. By Remark 2.5(c), it’s
sufficient to show that if G is $H\sqcup S_{2,t}$-free dense 3-graph then
$\lambda(G)\leq\lambda(K_{s+t+1}^{3})$. Suppose on the contrary that
$\lambda(G)>\lambda(K_{s+t+1}^{3})$. Let $\vec{x}$ be an optimal weight vector
of G.

_Case 1._ There exists $v\in V(G)$ with weight $x_{v}$ such that $G-\\{v\\}$
is H-free.

By Claim 3.1, $x_{v}<1-\frac{\sqrt{(s+t)(s+t-1)}}{s+t+1}$. By Claim 3.3 and
that H is $\lambda$-perfect,

$\lambda(G)\leq\frac{\lambda(K_{s-1}^{3})(1-x_{v})^{3}}{1-3x_{v}}=f(x_{v}).$
$None$

By Remark 3.4, $f(x_{v})$ is increasing in
$[0,1-\frac{\sqrt{(s+t)(s+t-1)}}{s+t+1})$, then

$\displaystyle\lambda(G)$ $\displaystyle\leq$ $\displaystyle
f(1-\frac{\sqrt{(s+t)(s+t-1)}}{s+t+1})$ $\displaystyle=$
$\displaystyle\frac{1}{6}\frac{(s-2)(s-3)}{(s-1)^{2}}\frac{\frac{(s+t)(s+t-1)\sqrt{(s+t)(s+t-1)}}{(s+t+1)^{3}}}{\frac{3\sqrt{(s+t)(s+t-1)}}{s+t+1}-2}$
$\displaystyle=$
$\displaystyle\frac{1}{6}\frac{(s+t)(s+t-1)}{(s+t+1)^{2}}\times\frac{(s-2)(s-3)}{(s-1)^{2}}\times\frac{\sqrt{(s+t)(s+t-1)}}{3\sqrt{(s+t)(s+t-1)}-2(s+t+1)}$
$\displaystyle=$
$\displaystyle\lambda(K_{s+t+1}^{3})\times\frac{(s-2)(s-3)}{(s-1)^{2}}\times\frac{\sqrt{(s+t)(s+t-1)}}{3\sqrt{(s+t)(s+t-1)}-2(s+t+1)}.$

To prove $\lambda(G)\leq\lambda(K_{s+t+1}^{3})$, it’s sufficient to prove that

$\frac{(s-2)(s-3)}{(s-1)^{2}}\times\frac{\sqrt{(s+t)(s+t-1)}}{3\sqrt{(s+t)(s+t-1)}-2(s+t+1)}\leq
1.$

This is equivalent to

$\frac{2(s-1)^{2}(s+t+1)}{2s^{2}-s-3}\leq\sqrt{(s+t)(s+t-1)}.$

The above inequality is equivalent to

$s+t-\frac{1}{2}-\frac{(6t-1)s-10t-1}{4s^{2}-2s-6}\leq\sqrt{(s+t)(s+t-1)}.$

By direct calculation, it holds for $s$=3 or 4 and $t=1$. Since

$s+t-\frac{1}{2}-\frac{(6t-1)s-10t-1}{4s^{2}-2s-6}\leq
s+t-\frac{1}{2}-\frac{1}{s+1},$

and

$s+t-\frac{1}{2}-\frac{1}{s+1}\leq\sqrt{(s+t)(s+t-1)}$

holds for any $s\geq 3$ and $t\geq 2$ or $s\geq 5$ and $t=1$. So
$\lambda(G)\leq\lambda(K_{s+t+1}^{3})$, a contradiction.

_Case 2._ For any $v\in V(G)$, $H\subseteq G-\\{v\\}$.

Since $\lambda(G)>\lambda(K_{s+t+1}^{3})$ and $S_{2,s+t}$ is
$\lambda$-perfect, then
$S_{2,s+t}=\\{v_{1}v_{2}b_{1},v_{1}v_{2}b_{2},...,v_{1}v_{2}b_{s+t}\\}\subseteq
G$. By Claim 3.5, $G-\\{v_{1},v_{2}\\}$ is H-free. By Claim 3.6, we have

$\omega(G_{v_{1}})\leq s+t\quad{\rm and}\quad\omega(G_{v_{2}})\leq s+t.$

By Claim 3.7, we have $\omega((G-\\{v_{2}\\})_{v_{1}})\leq s+t-1$ and
$\omega((G-\\{v_{1}\\})_{v_{2}})\leq s+t-1$.

Assume the weight of $v_{1}$ and $v_{2}$ are $a_{1}$ and $a_{2}$ respectively,
and $a_{1}+a_{2}=2a$. Since $G-\\{v_{1},v_{2}\\}$ is H-free and H is
$\lambda$-perfect, the contribution of edges containing neither $v_{1}$ nor
$v_{2}$ to $\lambda(G,\vec{x})$ is at most $\lambda(K_{s-1}^{3})(1-2a)^{3}$.
Since $\omega((G-\\{v_{2}\\})_{v_{1}})\leq s+t-1$ and
$\omega((G-\\{v_{1}\\})_{v_{2}})\leq s+t-1$, by Theorem 1.2, the contribution
of edges containing either $v_{1}$ or $v_{2}$ to $\lambda(G,\vec{x})$ is at
most $2\times\frac{1}{2}a(1-\frac{1}{s+t-1})(1-2a)^{2}$. The contribution of
edges containing both $v_{1}$ and $v_{2}$ to $\lambda(G,\vec{x})$ is at most
$a^{2}(1-2a)$. Therefore

$\displaystyle\lambda(G)$ $\displaystyle\leq$
$\displaystyle\lambda(K_{s-1}^{3})(1-2a)^{3}+a^{2}(1-2a)+2\times\frac{1}{2}a(1-\frac{1}{s+t-1})(1-2a)^{2}$
$\displaystyle\leq$
$\displaystyle\lambda(K_{s+t-1}^{3})(1-2a)^{3}+a^{2}(1-2a)+2\times\frac{1}{2}a(1-\frac{1}{s+t-1})(1-2a)^{2}$
$\displaystyle=$
$\displaystyle\lambda(K_{s+t+1}^{3},(a,a,\frac{1-2a}{s+t-1},...,\frac{1-2a}{s+t-1}))$
$\displaystyle\leq$ $\displaystyle\lambda(K_{s+t+1}^{3}).$

$\Box$

###  3.3 Disjoint union of $F_{5}$ and $S_{2,t}$

We show that the disjoint union of $F_{5}$ and $S_{2,t}$ is $\lambda$-perfect.

_Proof of Theorem 1.9._ Let $t\geq 2.$ Let G be a dense $F_{5}\sqcup
S_{2,t}$-free 3-graph on n vertices. By Remark 2.5(c), it’s sufficient to show
that $\lambda(G)\leq\lambda(K_{t+6}^{3})$. Suppose on the contrary that
$\lambda(G)>\lambda(K_{t+6}^{3})$. By Claim 3.1, for any $v\in V(G)$ with
weight $x_{v}$ we have

$x_{v}<1-\frac{\sqrt{(t+5)(t+4)}}{t+6}.$

_Case 1._ $G-\\{v\\}$ is $F_{5}$-free for some $v\in V(G)$.

By the result in [30] that $\pi_{\lambda}(F_{5})=\frac{4}{9}$ and Claim 3.3,
we have

$\displaystyle\lambda(G)$ $\displaystyle\leq$
$\displaystyle\frac{2}{27}\frac{(1-x_{v})^{3}}{1-3x_{v}}=f(x_{v})$
$\displaystyle\leq$ $\displaystyle f(1-\frac{\sqrt{(t+5)(t+4)}}{t+6})$
$\displaystyle=$
$\displaystyle\frac{1}{6}\frac{(t+4)(t+5)}{(t+6)^{2}}\times\frac{4}{9}\times\frac{\sqrt{(t+5)(t+4)}}{3\sqrt{(t+5)(t+4)}-2(t+6)}$
$\displaystyle=$
$\displaystyle\lambda(K_{t+6}^{3})\times\frac{4}{9}\times\frac{\sqrt{(t+5)(t+4)}}{3\sqrt{(t+5)(t+4)}-2(t+6)}.$

It’s sufficient to show that

$\frac{4}{9}\times\frac{\sqrt{(t+5)(t+4)}}{3\sqrt{(t+5)(t+4)}-2(t+6)}\leq 1.$

It’s equivalent to show that

$18(t+6)\leq 23\sqrt{(t+5)(t+4)}.$

It holds for $t\geq 2.$

_Case 2._ $G-\\{v\\}$ contains $F_{5}$ for any $v\in V(G)$.

Since $\lambda(G)>\lambda(K_{t+6}^{3})$ and $S_{2,t+5}$ is $\lambda$-perfect,
then
$S_{2,t+5}=\\{v_{1}v_{2}u_{1},v_{1}v_{2}u_{2},...,v_{1}v_{2}u_{t+5}\\}\subseteq
G$. Assume the weight of $v_{1}$ and $v_{2}$ are $a_{1}$ and $a_{2}$
respectively, and $a_{1}+a_{2}=2a$. By Claim 3.5, $G-\\{v_{1},v_{2}\\}$ is
$F_{5}$-free. So the contribution of edges containing neither $v_{1}$ nor
$v_{2}$ to $\lambda(G,\vec{x})$ is at most
$\frac{\pi_{\lambda}(F_{5})}{6}(1-2a)^{3}=\frac{2}{27}(1-2a)^{3}$. By Claim
3.7, we have $\omega((G-\\{v_{2}\\})_{v_{1}})\leq t+4$ and
$\omega((G-\\{v_{1}\\})_{v_{2}})\leq t+4$. By Theorem 1.2, the contribution of
edges containing either $v_{1}$ or $v_{2}$ to $\lambda(G,\vec{x})$ is at most
$2a\times\frac{1}{2}(1-\frac{1}{t+4})(1-2a)^{2}$. The contribution of edges
containing both $v_{1}$ and $v_{2}$ to $\lambda(G,\vec{x})$ is at most
$a^{2}(1-2a)$. Therefore

$\displaystyle\lambda(G)$ $\displaystyle\leq$
$\displaystyle\frac{2}{27}(1-2a)^{3}+a^{2}(1-2a)+2a\times\frac{1}{2}(1-\frac{1}{t+4})(1-2a)^{2}$
$\displaystyle\leq$
$\displaystyle\lambda(K_{t+4}^{3})(1-2a)^{3}+a^{2}(1-2a)+2a\times\frac{1}{2}(1-\frac{1}{t+4})(1-2a)^{2}$
$\displaystyle=$
$\displaystyle\lambda(K_{t+6}^{3},(a,a,\frac{1-2a}{t+4},...,\frac{1-2a}{t+4}))$
$\displaystyle\leq$ $\displaystyle\lambda(K_{t+6}^{3}).$

$\Box$

The following lemma is prepared for the proof of Theorem 1.8.

###### Lemma 3.9

Let G be a dense and $F_{5}\sqcup\\{e\\}$-free 3-graph. If $K_{4}^{3-}\sqcup
K_{4}^{3-}\subseteq G$, then $\lambda(G)<\frac{5}{49}=\lambda(K_{7}^{3}).$

_Proof._ Let $K=K_{4}^{3-}\sqcup K_{4}^{3-}$ with vertex set
$\\{a_{1},a_{2},b_{1},b_{2},...,b_{6}\\}$ and edge set
$\\{a_{1}b_{1}b_{2},a_{1}b_{1}b_{3},a_{1}b_{2}b_{3},a_{2}b_{4}b_{5},\\\
a_{2}b_{4}b_{6},a_{2}b_{5}b_{6}\\}$. Let $\vec{x}$ be an optimal weight
vector. To simplify the notation, we simply write the weight of vertex $a$ in
$\vec{x}$ as $a$. In other words, in the proof, $a$ sometimes means vertex
$a$, sometimes means the weight of vertex $a$ in the optimal weight vector
$\vec{x}$. By Fact 2.2,

$24\lambda=\sum_{i=1}^{2}\frac{\partial\lambda}{\partial
a_{i}}+\sum_{i=1}^{6}\frac{\partial\lambda}{\partial b_{i}}.$ $None$

###### Claim 3.10

i) For vertices $c_{i},c_{j}\in V(G)\setminus V(K)$, the product $c_{i}c_{j}$
appears at most twice in $\sum_{i=1}^{2}\frac{\partial\lambda}{\partial
a_{i}}+\sum_{i=1}^{6}\frac{\partial\lambda}{\partial b_{i}}$.

ii) For vertices $c_{i}\in V(G)\setminus V(K)$, the product $c_{i}b_{j}$
appears at most twice in $\sum_{i=1}^{2}\frac{\partial\lambda}{\partial
a_{i}}+\sum_{i=1}^{6}\frac{\partial\lambda}{\partial b_{i}}$.

iii) For vertices $c_{i}\in V(G)\setminus V(K)$, the product $c_{i}a_{1}$,
$c_{i}a_{2}$ appear at most 7 times in
$\sum_{i=1}^{2}\frac{\partial\lambda}{\partial
a_{i}}+\sum_{i=1}^{6}\frac{\partial\lambda}{\partial b_{i}}$.

iv) The product $b_{i}b_{j}$ appears at most 3 times in
$\sum_{i=1}^{2}\frac{\partial\lambda}{\partial
a_{i}}+\sum_{i=1}^{6}\frac{\partial\lambda}{\partial b_{i}}$.

v) The product $a_{i}b_{j}$, $a_{1}a_{2}$ appear at most 6 times in
$\sum_{i=1}^{2}\frac{\partial\lambda}{\partial
a_{i}}+\sum_{i=1}^{6}\frac{\partial\lambda}{\partial b_{i}}$.

_Proof._ i)Let F be a copy of $F_{5}\cup\\{e\\}$. We first show that
$c_{i}c_{j}$ can appear at most 1 time in $\frac{\partial\lambda}{\partial
a_{1}}+\sum_{i=1}^{3}\frac{\partial\lambda}{\partial b_{i}}$. If not, suppose
that $c_{i}c_{j}$ appears at least 2 times in it. If
$c_{i}c_{j}b_{s},c_{i}c_{j}b_{t}\in E(G)$ where
$b_{s},b_{t}\in\\{b_{1},b_{2},b_{3}\\}$, then
$\\{a_{1},b_{s},b_{t},c_{i},c_{j}\\}$ forms an $F_{5}$. So there is an $F$
formed by $F_{5}\sqcup\\{a_{2}b_{4}b_{5}\\}$. If
$c_{i}c_{j}b_{s},c_{i}c_{j}a_{1}\in E(G)$ where
$b_{s}\in\\{b_{1},b_{2},b_{3}\\}$, then $\\{a_{1},b_{t},b_{s},c_{i},c_{j}\\}$
forms an $F_{5}$, where $b_{t}\in\\{b_{1},b_{2},b_{3}\\}\setminus\\{b_{s}\\}$.
So $F_{5}\sqcup\\{a_{2}b_{4}b_{5}\\}\subseteq G$. Similarly, $c_{i}c_{j}$ can
appear at most 1 time in $\frac{\partial\lambda}{\partial
a_{2}}+\sum_{i=4}^{6}\frac{\partial\lambda}{\partial b_{i}}$. Therefore
$c_{i}c_{j}$ appears at most twice in
$\sum_{i=1}^{2}\frac{\partial\lambda}{\partial
a_{i}}+\sum_{i=1}^{6}\frac{\partial\lambda}{\partial b_{i}}$.

ii) Without loss of generality, let $b_{j}=b_{1}$. We first show that
$c_{i}b_{1}$ appears in $\frac{\partial\lambda}{\partial
a_{1}}+\frac{\partial\lambda}{\partial b_{2}}+\frac{\partial\lambda}{\partial
b_{3}}$ at most 1 time. Since if $c_{i}b_{1}b_{2},c_{i}b_{1}b_{3}\in E(G)$,
then $\\{a_{1},b_{1},b_{2},b_{3},c_{i}\\}$ forms an $F_{5}$, then there is an
$F$ formed by $F_{5}\sqcup\\{a_{2}b_{4}b_{5}\\}$. If $c_{i}b_{1}b_{2}$ (or
$c_{i}b_{1}b_{3}$) and $c_{i}b_{1}a_{1}\in E(G)$, then
$\\{a_{1},b_{1},b_{2},b_{3},c_{i}\\}$ forms an $F_{5}$, then there is an $F$
formed by $F_{5}\sqcup\\{a_{2}b_{4}b_{5}\\}$.

Next we show that $c_{i}b_{1}$ appears in $\frac{\partial\lambda}{\partial
a_{2}}+\sum_{i=4}^{6}\frac{\partial\lambda}{\partial b_{i}}$ at most 1 time.
If $c_{i}b_{1}b_{s},c_{i}b_{1}b_{t}\in E(G)$, where
$b_{s},b_{t}\in\\{b_{4},b_{5},b_{6}\\}$, then
$\\{a_{2},b_{1},b_{s},b_{t},c_{i}\\}$ forms an $F_{5}$, and there is an $F$
formed by $F_{5}\sqcup\\{a_{1}b_{2}b_{3}\\}$. If
$c_{i}b_{1}a_{2},c_{i}b_{1}b_{s}\in E(G)$, where
$b_{s}\in\\{b_{4},b_{5},b_{6}\\}$, then $\\{a_{2},b_{1},b_{s},b_{t},c_{i}\\}$
forms an $F_{5}$ where $b_{s},b_{t}\in\\{b_{4},b_{5},b_{6}\\}$. So there is an
$F$ formed by $F_{5}\sqcup\\{a_{1}b_{2}b_{3}\\}$.

iii) Let $j=1$ or 2. Since $c_{i}a_{j}\notin\frac{\partial\lambda}{\partial
a_{j}}$, then $c_{i}a_{j}$ appears at most 7 times.

iv) If $b_{i}\in\\{b_{1},b_{2},b_{3}\\}$ and
$b_{j}\in\\{b_{4},b_{5},b_{6}\\}$, without loss of generality, let $i=1$ and
$j=4$. Note that $b_{1}b_{2}b_{4}\notin E(G)$. Otherwise
$\\{a_{1},b_{1},b_{2},b_{3},b_{4}\\}$ forms an $F_{5}$ and there is an $F$
formed by $F_{5}\sqcup\\{a_{2}b_{5}b_{6}\\}$. Similarly,
$b_{1}b_{3}b_{4},b_{1}b_{4}b_{5},b_{1}b_{4}b_{6},b_{1}b_{2}b_{5},b_{1}b_{2}b_{6}\notin
E(G)$. So at most $b_{1}b_{4}a_{1},b_{1}b_{4}a_{2}\in E(G)$. If
$b_{i},b_{j}\in\\{b_{1},b_{2},b_{3}\\}$, without loss of generality, let $i=1$
and $j=2$. Then at most $b_{1}b_{2}a_{1},b_{1}b_{2}a_{2},b_{1}b_{2}b_{3}\in
E(G)$.

v) Since $a_{i}b_{j}$ can not appear in $\frac{\partial\lambda}{\partial
a_{j}}+\frac{\partial\lambda}{\partial b_{i}}$, then $a_{i}b_{j}$ can appear
at most 6 times in $\sum_{i=1}^{2}\frac{\partial\lambda}{\partial
a_{i}}+\sum_{i=1}^{6}\frac{\partial\lambda}{\partial b_{i}}$. The reason for
$a_{1}a_{2}$ is similar. $\Box$

Combining (1) and Claim 3.3, we have

$24\lambda\leq 2\sum_{c_{i},c_{j}\in V(G)\setminus
V(K)}c_{i}c_{j}+2\sum_{i\in[6],c_{j}\in V(G)\setminus
V(K)}b_{i}c_{j}+7\sum_{i\in[2],c_{j}\in V(G)\setminus V(K)}a_{i}c_{j}+$
$3\sum_{i\neq j\in[6]}b_{i}b_{j}+\\\
6\sum_{i=1}^{2}\sum_{j=1}^{6}a_{i}b_{j}+6a_{1}a_{2}.$

Let $\sum_{i}c_{i}=c$, $\sum_{i=1}^{2}a_{i}=a$, $\sum_{i=1}^{6}b_{i}=b$, then
$a+b+c=1$. So

$\displaystyle 24\lambda$ $\displaystyle\leq$ $\displaystyle
2\bigg{(}\frac{c}{n-8}\bigg{)}^{2}\binom{n-8}{2}+2bc+7ac+3\bigg{(}\frac{b}{6}\bigg{)}^{2}\binom{6}{2}+6ab+6\bigg{(}\frac{a}{2}\bigg{)}^{2}$
$\displaystyle\leq$ $\displaystyle
c^{2}+2bc+7ac+\frac{3}{2}b^{2}+6ab+\frac{3}{2}a^{2}$ $\displaystyle=$
$\displaystyle(1-a-b)^{2}+2b(1-a-b)+7a(1-a-b)+\frac{3}{2}b^{2}+6ab+\frac{3}{2}a^{2}$
$\displaystyle=$ $\displaystyle-\frac{9}{2}a^{2}+\frac{1}{2}b^{2}-ab+5a+1$
$\displaystyle=$ $\displaystyle-\frac{9}{2}a^{2}+(5-b)a+\frac{1}{2}b^{2}+1$
$\displaystyle\leq$
$\displaystyle\frac{5}{9}b^{2}-\frac{5}{9}b+\frac{43}{18}\qquad(a=\frac{5-b}{9})$
$\displaystyle\leq$ $\displaystyle\frac{43}{18}.$

So

$\lambda(G)\leq\frac{43}{18\times 24}<\frac{5}{49}=\lambda(K_{7}^{3}).$

$\Box$
_Proof of Theorem 1.8._ Let G be a dense $F_{5}\cup\\{e\\}$-free 3-graph. By
Remark 2.5(c), it’s sufficient to show that
$\lambda(G)\leq\lambda(K_{7}^{3})$. Suppose on the contrary that
$\lambda(G)>\frac{5}{49}=\lambda(K_{7}^{3})$. For $v\in V(G)$ with weight
$x_{v}$, if $G-\\{v\\}$ is $K_{4}^{3-}$-free, then by Proposition 1.3, Theorem
3.3 and Claim 3.8, we have

$\lambda(G)\leq\frac{\pi_{\lambda}(K_{4}^{3-})}{6}\frac{(1-x_{v})^{3}}{1-3x_{v}}=\frac{\pi(K_{4}^{3-})}{6}\frac{(1-x_{v})^{3}}{1-3x_{v}}\leq\frac{1}{18}\frac{(1-x_{v})^{3}}{1-3x_{v}}.$

By Claim 3.1, we have $x_{v}<1-\frac{\sqrt{30}}{7}<\frac{1}{3}.$ By Remark
3.4, $\frac{1}{18}\frac{(1-x_{v})^{3}}{1-3x_{v}}$ is increasing in
$(0,\frac{1}{3})$, so

$\lambda(G)\leq\frac{1}{18}\frac{(1-x_{v})^{3}}{1-3x_{v}}\bigg{|}_{x_{v}=1-\frac{\sqrt{30}}{7}}<\frac{5}{49}.$

Hence we may assume that $G-\\{v\\}$ still contains $K_{4}^{3-}$ for any
vertex $v$. Note that $\omega(G_{v})\leq 6$. Since if $\omega(G_{v})\geq 7$,
assume that $U=\\{u_{1},u_{2},...,u_{7}\\}$ is a clique in $G_{v}$. Let
$K_{4}^{3-}\subseteq G-\\{v\\}$ have the vertex set W. Then $|U-W|\geq 3$ and
we can find a $K_{4}^{3-}$ in $\\{v_{1}\\}\cup U-W$. Therefore
$K_{4}^{3-}\sqcup K_{4}^{3-}\subseteq G$. So by Fact 2.2 and the theorem of
Motzkin and Straus (Theorem 1.2), we have

$3\times\frac{5}{49}<3\lambda=\frac{\partial\lambda}{\partial
x_{v}}\leq\frac{1}{2}(1-\frac{1}{6})(1-x_{v})^{2}.$

So $x_{v}<\frac{1}{7}$ holds for any $v$.

If for $v\in V(G)$ with weight $x_{v}$, $G-\\{v\\}$ is $F_{5}$-free, then by
the result in [30] that $\pi_{\lambda}(F_{5})=\frac{4}{9}$ and Claim 3.3, we
have

$\lambda(G)\leq\frac{2}{27}\frac{(1-x_{v})^{3}}{1-3x_{v}}\leq\frac{2}{27}\frac{(1-x_{v})^{3}}{1-3x_{v}}\bigg{|}_{x_{v}=\frac{1}{7}}<\frac{5}{49}.$

So we may assume that $G-\\{v\\}$ still contains both $F_{5}$ and $K_{4}^{3-}$
for any vertex $v$. Since $\lambda(G)>\frac{5}{49}$ and $S_{2,t}$ is
$\lambda$-perfect, then $S_{2,6}\subseteq G.$ Let
$S_{2,6}=\\{v_{1}v_{2}b_{1},v_{1}v_{2}b_{2},...,v_{1}v_{2}b_{6}\\}$. By Claim
3.5, $G-\\{v_{1},v_{2}\\}$ is $F_{5}$-free. By Claim 3.7,
$\omega((G-\\{v_{2}\\})_{v_{1}})\leq 5$ and
$\omega((G-\\{v_{1}\\})_{v_{2}})\leq 5$. Assume the weight of $v_{1}$ and
$v_{2}$ are $a_{1}$ and $a_{2}$ respectively and $a_{1}+a_{2}=2a$. By Claim
3.2, $a_{1}<\frac{1}{7}$ and $a_{1}<\frac{1}{7}$. Therefore $a<\frac{1}{7}$.
So

$\displaystyle\lambda(G)$ $\displaystyle\leq$ $\displaystyle
a^{2}(1-2a)+\frac{2}{27}(1-2a)^{3}+2a\times\frac{1}{2}\times(1-\frac{1}{5})(1-2a)^{2}$
$\displaystyle=$
$\displaystyle\frac{82}{135}a^{3}-\frac{59}{45}a^{2}+\frac{16}{45}a+\frac{2}{27}=f(a)$
$\displaystyle f^{\prime}(a)$ $\displaystyle=$
$\displaystyle\frac{82}{45}a^{2}-\frac{118}{45}a+\frac{16}{45}>0$

if $a\in[0,\frac{1}{7}]$. So $f(a)$ is increasing in $[0,\frac{1}{7}]$. Then

$\lambda(G)\leq f(a)=f(\frac{1}{7})<\frac{5}{49}.$

$\Box$

###  3.4 The disjoint union of $K_{4}^{3-}$ and $S_{2,t}$

We show that the disjoint union of $K_{4}^{3-}$ and $S_{2,t}$ is
$\lambda$-perfect.

_Proof of Theorem 1.10._ Let G be a dense $K_{4}^{3-}\sqcup S_{2,t}$-free
3-graph on n vertices. By Remark 2.5(c), it’s sufficient to show that
$\lambda(G)\leq\lambda(K_{t+5}^{3})$. Suppose on the contrary that
$\lambda(G)>\lambda(K_{t+5}^{3})$. If there exists $v\in V(G)$ with weight
$x_{v}$ such that $G-\\{v\\}$ is $S_{2,t}$-free, then by Claim 3.1, Claim 3.3
and $S_{2,t}$ is $\lambda$-perfect, we have

$x_{v}<1-\frac{\sqrt{(t+3)(t+4)}}{t+5}\quad{\rm
and}\quad\lambda(G)\leq\frac{\pi_{\lambda}(S_{2,t})(1-x_{v})^{3}}{6(1-3x_{v})}\leq\frac{\lambda(K_{t+1}^{3})(1-x_{v})^{3}}{1-3x_{v}}=f(x_{v}).$

Since $f(x)$ is increasing in $(0,\frac{1}{3})$, then

$\displaystyle\lambda(G)$ $\displaystyle\leq$ $\displaystyle
f(1-\frac{\sqrt{(t+3)(t+4)}}{t+5})$ $\displaystyle=$
$\displaystyle\frac{1}{6}\frac{t(t-1)}{(t+1)^{2}}\frac{\frac{(t+4)(t+3)\sqrt{(t+4)(t+3)}}{(t+5)^{3}}}{\frac{3\sqrt{(t+4)(t+3)}}{t+5}-2}$
$\displaystyle=$
$\displaystyle\frac{1}{6}\frac{(t+4)(t+3)}{(t+5)^{2}}\frac{t(t-1)}{(t+1)^{2}}\frac{\sqrt{(t+4)(t+3)}}{3\sqrt{(t+4)(t+3)}-2(t+5)}$
$\displaystyle=$
$\displaystyle\lambda(K_{t+5}^{3})\frac{t(t-1)}{(t+1)^{2}}\frac{\sqrt{(t+4)(t+3)}}{3\sqrt{(t+4)(t+3)}-2(t+5)}.$

So it’s sufficient to show that

$\frac{t(t-1)}{(t+1)^{2}}\times\frac{\sqrt{(t+4)(t+3)}}{3\sqrt{(t+4)(t+3)}-2(t+5)}\leq
1.$

This is equivalent to show that

$t+\frac{7}{2}-\frac{11t+1}{4t^{2}+14t+6}\leq\sqrt{(t+4)(t+3)}.$

This is true since

$\frac{11t+1}{4t^{2}+14t+6}\geq\frac{1}{t+1},(t\geq 1)$

and

$t+\frac{7}{2}-\sqrt{(t+4)(t+3)}=\frac{(t+\frac{7}{2})^{2}-(t+3)(t+4)}{t+\frac{7}{2}+\sqrt{(t+4)(t+3)}}<\frac{1}{4(t+1)}<\frac{1}{t+1}.$

So we may assume that $S_{2,t}\in G-\\{v\\}$ for any $v\in V(G)$. By Claim
3.6, $\omega(G_{v})\leq t+4$. Let $x_{v}$ be the weight of v in an optimum
vector. By Claim 3.2, $x_{v}<\frac{1}{t+5}$.

_Case 1._ $G-\\{v\\}$ is $K_{4}^{3-}$-free for some $v\in V(G)$.

Then by Proposition 1.3, Claim 3.3 and Theorem 3.8, we have

$\displaystyle\lambda(G)$ $\displaystyle\leq$
$\displaystyle\frac{1}{18}\frac{(1-x_{v})^{3}}{1-3x_{v}}=f(x_{v})\quad(x_{v}<\frac{1}{t+5})$
$\displaystyle<$ $\displaystyle f(\frac{1}{t+5})$ $\displaystyle=$
$\displaystyle\frac{1}{18}\frac{(t+4)^{3}}{(t+2)(t+5)^{2}}$ $\displaystyle<$
$\displaystyle\frac{1}{6}\frac{(t+4)(t+3)}{(t+5)^{2}}=\lambda(K_{t+5}^{3}).$

_Case 2._ $G-\\{v\\}$ contains $K_{4}^{3-}$ for any $v\in V(G)$.

Since $\lambda(G)>\lambda(K_{t+5}^{3})$ and $S_{2,t+4}$ is $\lambda$-perfect,
then
$S_{2,t+4}=\\{v_{1}v_{2}u_{1},v_{1}v_{2}u_{2},...,v_{1}v_{2}u_{t+4}\\}\subseteq
G$. By Claim 3.5, $G-\\{v_{1},v_{2}\\}$ is $K_{4}^{3-}$-free. Since
$G-\\{v_{1}\\}$ contains $K_{4}^{3-}$ and $S_{2,t}$, by Claim 3.7,
$\omega((G-\\{v_{2}\\})_{v_{1}})\leq t+3$ and
$\omega((G-\\{v_{1}\\})_{v_{2}})\leq t+3$. Assume the weight of $v_{1}$ and
$v_{2}$ are $a_{1}$ and $a_{2}$ respectively. Let $a_{1}+a_{2}=2a$. So

$\displaystyle\lambda(G)$ $\displaystyle\leq$
$\displaystyle\frac{1}{18}(1-2a)^{3}+a^{2}(1-2a)+2a\times\frac{1}{2}(1-\frac{1}{t+3})(1-2a)^{2}$
$\displaystyle\leq$
$\displaystyle\lambda(K_{t+3}^{3})(1-2a)^{3}+a^{2}(1-2a)+2a\times\frac{1}{2}(1-\frac{1}{t+3})(1-2a)^{2}$
$\displaystyle<$
$\displaystyle\lambda(K_{t+5}^{3},(a,a,\frac{1-2a}{t+3},...,\frac{1-2a}{t+3}))$
$\displaystyle\leq$ $\displaystyle\lambda(K_{t+5}^{3}).$

$\Box$

###  3.5 General case

We show that $H\sqcup S_{2,t}$ is $\lambda$-perfect if H is a $3$-graph with
$s$ vertices and $t\geq\frac{3}{2}s^{2}-\frac{11}{2}s+4$.

_Proof of Theorem 1.11._ Let G be a dense $H\sqcup S_{2,t}$-free 3-graph on n
vertices. By Remark 2.5(c), it’s sufficient to show that
$\lambda(G)\leq\lambda(K_{s+t+1}^{3})$. Suppose on the contrary that
$\lambda(G)>\lambda(K_{s+t+1}^{3})$. If there exists $v\in V(G)$ with weight
$x_{v}$ such that $G-\\{v\\}$ is $H$-free, then by Claim 3.1 and Claim 3.3, we
have

$\lambda(G)\leq\frac{1}{6}\pi_{\lambda}(H)\frac{(1-x_{v})^{3}}{1-3x_{v}},\quad{\rm
and}\quad x_{v}<1-\frac{\sqrt{(s+t)(s+t-1)}}{s+t+1}.$

Since $\pi_{\lambda}(H)\leq\pi_{\lambda}(K_{s}^{3})\leq
1-\frac{2}{(s-1)(s-2)}$ (see [15]), then

$\displaystyle\lambda(G)$ $\displaystyle\leq$
$\displaystyle\frac{1}{6}\bigg{(}1-\frac{2}{(s-1)(s-2)}\bigg{)}\frac{(1-x_{v})^{3}}{1-3x_{v}}\bigg{|}_{x_{v}=1-\frac{\sqrt{(s+t)(s+t-1)}}{s+t+1}}$
$\displaystyle\leq$
$\displaystyle\lambda(K_{s+t+1}^{3})\frac{s^{2}-3s}{(s-1)(s-2)}\times\frac{\sqrt{(s+t)(s+t-1)}}{3\sqrt{(s+t)(s+t-1)}-2}.$

So it’s sufficient to show that

$\frac{s^{2}-3s}{(s-1)(s-2)}\times\frac{\sqrt{(s+t)(s+t-1)}}{3\sqrt{(s+t)(s+t-1)}-2}\leq
1.$

This is equivalent to show that

$1-\frac{1}{s^{2}-3s+3}\leq\sqrt{(s+t)(s+t-1)}.$

It holds for $s\geq 3$ and $t\geq 1.$

So we can assume that $H\subseteq G-\\{v\\}$ for any $v\in V(G)$. By Claim 3.6
and Claim 3.2, we have $\omega(G_{v})\leq s+t$ and

$x_{v}<\frac{1}{s+t+1}.$

Since $\lambda(G)>\lambda(K_{s+t+1}^{3})$ and $S_{2,s+t}$ is
$\lambda$-perfect, then
$S_{2,s+t}=\\{v_{1}v_{2}u_{1},v_{1}v_{2}u_{2},...,v_{1}v_{2}u_{s+t}\\}\subseteq
G$. By Claim 3.5, $G-\\{v_{1},v_{2}\\}$ is $H$-free. Applying Claim 3.7, we
have $\omega((G-\\{v_{2}\\})_{v_{1}})\leq s+t-1$ and
$\omega((G-\\{v_{1}\\})_{v_{2}})\leq s+t-1$. Since
$t\geq\frac{3}{2}s^{2}-\frac{11}{2}s+4$, then

$6\lambda(K_{s+t-1}^{3})=\frac{(s+t-2)(s+t-3)}{(s+t-1)^{2}}\geq\frac{s^{2}-3s}{s^{2}-3s+2}>\pi_{\lambda}(H).$

Assume the weight of $v_{1}$ and $v_{2}$ are $a_{1}$ and $a_{2}$ respectively,
and $a_{1}+a_{2}=2a$, then

$\displaystyle\lambda(G)$ $\displaystyle\leq$
$\displaystyle\frac{1}{6}\pi_{\lambda}(H)(1-2a)^{3}+a^{2}(1-2a)+2a\times\frac{1}{2}(1-\frac{1}{s+t-1})(1-2a)^{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{6}\frac{s^{2}-3s}{(s-1)(s-2)}(1-2a)^{3}+a^{2}(1-2a)+2a\times\frac{1}{2}(1-\frac{1}{s+t-1})(1-2a)^{2}$
$\displaystyle\leq$
$\displaystyle\lambda(K_{s+t-1}^{3})(1-2a)^{3}+a^{2}(1-2a)+2a\times\frac{1}{2}(1-\frac{1}{s+t-1})(1-2a)^{2}$
$\displaystyle=$
$\displaystyle\lambda(K_{s+t+1}^{3},(a,a,\frac{1-2a}{s+t-1},...,\frac{1-2a}{s+t-1}))$
$\displaystyle\leq$ $\displaystyle\lambda(K_{s+t+1}^{3}).$

$\Box$

##  4 Irrational Lagrangian densities

In this section, we prove Theorem 1.14.

###### Fact 4.1

Let $V(S_{2}(n))$=[n] and
$E(S_{2}(n))$=$\\{12i|i\in[n]\setminus\\{1,2\\}\\}\cup\\{ijk|i\in[2],j,k\in[n]\setminus\\{1,2\\}\\}$,
then $\lambda(S_{2}(n))=\frac{\sqrt{3}}{18}$.

_Proof._ Let $\vec{x}=\\{x_{1},x_{2},...,x_{n}\\}$ be an optimal weight vector
on $S_{2}(n)$. By Fact 2.3, we may assume that $x_{1}=x_{2}$,
$x_{3}=x_{4}=...=x_{n}.$ Let $x_{1}=a$, then $x_{3}=\frac{1-2a}{n-2}$ and
$a<\frac{1}{2}$. So

$\displaystyle\lambda(S_{2}(n))$ $\displaystyle=$ $\displaystyle
a^{2}(1-2a)+2a\bigg{(}\frac{1-2a}{n-2}\bigg{)}^{2}\binom{n-2}{2}$
$\displaystyle\leq$ $\displaystyle 2a^{3}-3a^{2}+a=f(a)$ $\displaystyle\leq$
$\displaystyle f(\frac{3-\sqrt{3}}{6})=\frac{\sqrt{3}}{18}.$

$\Box$
_Proof of Theorem 1.14._ Since $S_{2}(n)$ is $\mathscr{P}$-free, then by Fact
4.1 we have $\pi_{\lambda}(\mathscr{P})\geq
3!\lambda(S_{2}(n))=\frac{\sqrt{3}}{3}$. For the upper bound, we assume that G
is a $\mathscr{P}$-free dense 3-graph with vertex set $[n]$. It’s sufficient
to show that $\lambda(G)\leq\frac{\sqrt{3}}{18}$. Suppose on the contrary that
$\lambda(G)>\frac{\sqrt{3}}{18}$, then $n>6$, since
$\lambda(K_{6}^{3})=\frac{5}{54}<\frac{\sqrt{3}}{18}$. By Proposition 1.3 and
Theorem 3.8, $\pi_{\lambda}(K_{4}^{3})=\pi(K_{4}^{3})<\frac{\sqrt{3}}{3}$. So
$K_{4}^{3}\subseteq G$, without loss of generality, let {1, 2, 3, 4} form a
$K_{4}^{3}$. Let $P=\\{123,124,134,234,561,562\\}$.

###### Lemma 4.2

$P\nsubseteq G$.

_Proof of Lemma 4.2._ Suppose that $P\subseteq G$. We claim that for any
$u_{1},u_{2}\in V(G)\setminus\\{1,2,3,4,5,6\\}$,
$N(u_{1},u_{2})\subseteq\\{1,2\\}$. If there exists $u_{3}\in
V(G)\setminus\\{1,2,3,4,u_{1},u_{2}\\}$ such that $u_{1}u_{2}u_{3}\in E(G)$,
then $\\{1,2,3,4,u_{1},u_{2},u_{3}\\}$ forms a $P_{1}$. If there exist
$u_{3}\in\\{3,4\\}$ such that $u_{1}u_{2}u_{3}\in E(G)$, then
$\\{1,2,3,4,5,6,u_{1},\\\ u_{2},u_{3}\\}$ forms a $P_{2}$. And $u_{1}34\notin
E(G)$ since otherwise $\\{1,2,3,4,5,6,u_{1}\\}$ forms a $P_{3}$. And
$u_{1}35\notin E(G)$ since otherwise $\\{1,2,3,4,5,6,u_{1}\\}$ forms a
$P_{4}$, similarly $u_{1}36,u_{1}45,u_{1}46\notin E(G)$. Since G is
$P_{1}$-free, then every edge contains a vertex from $\\{1,2,3,4\\}$. So we
have shown that $G\subseteq S_{2}(n)$. By Fact 4.1,
$\lambda(G)\leq\frac{\sqrt{3}}{18}$, a contradiction. $\Box$

For any $v\in V(G)$ with weight $x_{v}$, we claim that $\omega(G_{v})\geq 3$.
Otherwise by Theorem 1.2 and Fact 2.2, we have

$3\lambda=\frac{\partial\lambda}{\partial
x_{v}}\leq\frac{1}{2}(1-\frac{1}{2})(1-x_{v})^{2}\leq\frac{1}{4}<\frac{\sqrt{3}}{6},$

a contradiction. Let $x\in V(G)\setminus\\{1,2,3,4\\}$. Since
$\omega(G_{x})\geq 3$, then $x$ is contained in a $K_{4}^{3-}$, denoted by
$K_{x}$, and $d_{K_{x}}(x)=3$. Since G is $P_{1}$-free, then
$|K_{x}\cap\\{1,2,3,4\\}|$=2 or 3.

_Case 1._ There exists $x\in V(G)\setminus\\{1,2,3,4\\}$ such that
$|K_{x}\cap\\{1,2,3,4\\}|$=2.

Without loss of generality, let $K_{x}$=$\\{1,2,x,y\\}$. Then
$\\{1,2,3,4,x,y\\}$ forms a $P$, by Lemma 4.2, a contradiction.

_Case 2._ For any $x\in V(G)\setminus\\{1,2,3,4\\}$,
$|K_{x}\cap\\{1,2,3,4\\}|$=3.

Without loss of generality, let $K_{x}$=$\\{1,2,3,x\\}$ for some $x\in
V(G)\setminus\\{1,2,3,4\\}$. Then $K_{x}$ forms a $K_{4}^{3}$. We claim that
for any $y\in V(G)\setminus\\{1,2,3,4,x\\}$, $K_{y}$=$\\{1,2,3,y\\}$. If not,
without loss of generality, let $K_{y}$=$\\{2,3,4,y\\}$. Then
$K_{x}\cup\\{y42,y43\\}$ forms a $P$, by Lemma 4.2, a contradiction. We claim
that for any $v_{1},v_{2},v_{3}\in V(G)\setminus\\{1,2,3\\}$,
$v_{1}v_{2}v_{3}\notin E(G)$ since otherwise there exist a $P_{1}$. And for
any pair $v_{1},v_{2}\in V(G)\setminus\\{1,2,3\\}$, we claim that
$|N(v_{1},v_{2})\cap\\{1,2,3\\}|$=1. If $N(v_{1},v_{2})\cap\\{1,2,3\\}\geq 2$,
take $v_{3}\in V(G)\setminus\\{1,2,3,v_{1},v_{2}\\}$, then
$\\{1,2,3,v_{1},v_{2},v_{3}\\}$ forms a $P$, by Lemma 4.2, a contradiction.

Assume the weight of $i\in V(G)$ is $x_{i}$ and $a$=$x_{1}+x_{2}+x_{3}$.
Consider $\sum_{k=1}^{3}\frac{\partial\lambda}{\partial x_{k}}$. From the
previous discussion we know that $x_{i}x_{j}$ only appears once in
$\sum_{k=1}^{3}\frac{\partial\lambda}{\partial x_{k}}$ for $4\leq i,j\leq n$.

So

$\displaystyle 9\lambda$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{3}\frac{\partial\lambda}{\partial x_{k}}$
$\displaystyle\leq$ $\displaystyle\bigg{(}\sum_{i,j\in
V(G)\setminus\\{1,2,3\\}}x_{i}x_{j}\bigg{)}+x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}+2(x_{1}+x_{2}+x_{3})\sum_{j\geq
4}x_{j}$ $\displaystyle=$
$\displaystyle\sum_{i,j}x_{i}x_{j}+(x_{1}+x_{2}+x_{3})\sum_{j\geq 4}x_{j}$
$\displaystyle\leq$ $\displaystyle\frac{1}{2}+a(1-a)\leq\frac{3}{4}.$

Then $\lambda(G)\leq\frac{1}{12}<\frac{\sqrt{3}}{18}$, a contradiction. $\Box$

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