# Strong field ionisation of Argon: Electron momentum spectra and nondipole
effects

Nida Haram<EMAIL_ADDRESS>Centre for Quantum Dynamics,
Griffith University, Brisbane, Queensland 4111, Australia  Han Xu  Centre for
Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia
Igor Ivanov  Centre for Relativistic Laser Science, Institute for Basic
Science, Gwangju 61005, Republic of Korea  D. Chetty  Centre for Quantum
Dynamics, Griffith University, Brisbane, Queensland 4111, Australia  Igor
Litvinyuk<EMAIL_ADDRESS>Centre for Quantum Dynamics, Griffith
University, Brisbane, Queensland 4111, Australia  R.T. Sang
<EMAIL_ADDRESS>Centre for Quantum Dynamics, Griffith University,
Brisbane, Queensland 4111, Australia

###### Abstract

We investigate the influence of relativistic nondipole effects on the
photoelectron spectra of argon, particularly in the low kinetic energy region
(0 eV - 5 eV). In our experiment, we use intense linearly polarised 800 nm
laser pulse to ionise Ar from a jet and we record photoelectron energy and
momentum distributions using a reaction microscope (REMI). Our measurements
show that nondipole effect can cause an energy dependent asymmetry along the
laser propagation direction in the photoelectron energy and momentum spectra.
Model simulation based on time-dependent Dirac equation (TDDE) can reproduce
our measurement results. The electron trajectory analysis based on classical
model reveals that the photoelectron which obtains negative momentum shift
along laser propagation direction is caused by the interplay between the
Lorenz force induced radiation pressure during its free propagation in
continuum and re-scattering by Coulomb potential of the parent ion when it is
driven back by the laser field.

##  I Introduction

In strong field physics, simple man’s three step model Corkum (1993a) has been
proven to be extremely useful for providing quantitative explanations to a
wealth of highly nonlinear laser-matter interaction phenomena, such as high-
order harmonic generation McPherson  _et al._ (1987); Ferray  _et al._ (1988);
Walser  _et al._ (2000), above-threshold ionisation (ATI) Corkum  _et al._
(1989); Eberly  _et al._ (1991); Milošević  _et al._ (2006), non-sequenctial
double ionisation Fittinghoff  _et al._ (1992); Walker  _et al._ (1994);
Feuerstein  _et al._ (2001), frustrated tunneling ionisation Nubbemeyer  _et
al._ (2008), etc. In this semiclassical model, the laser-matter interaction
happens in two main stages. At the first stage, the target atom or molecule is
tunnel ionised and a free electron appears in the continuum. At the second
stage, the free electron is driven solely by laser’s electric field while the
influence of the Coulomb potential is neglected.

The predictions of simple man’s model, however, could not show satisfactory
quantitative agreement with the results of high-resolution strong field
experiments. For example, the existence of low energy structure (LES), very
low energy structure and near zero energy structures in the photoelectron
energy and momentum spectra obtained as a result of strong field ionisation
with low frequency fields Blaga  _et al._ (2012); Faisal (2009); Wu  _et al._
(2012); Quan  _et al._ (2009a); Dura  _et al._ (2013); Pullen  _et al._
(2014); Wolter  _et al._ (2014, 2015) that arise due to the short-range
Coulomb focusing effect, which is an interaction that is ignored in simple
man’s model. Such short-range Coulomb focusing effects can only be observed
when the free electron is driven back to its parent ion core with a distance
from several tens to 100 atomic units with nearly zero velocity. Since the
Coulomb force scales inversely to the distance from the core (U $\sim$ 1/r),
the long-range Coulomb effect is supposed to be much weaker. The long-range
Coulomb effect has been found to significantly affect the emission angle of
the streaked photoelectrons Willenberg  _et al._ (2019) and strongly influence
the zero energy structure of the photoelectron momentum spectrum Quan  _et
al._ (2016).

The suppression of LES in case of experiment with circularly polarised laser
field led to the speculation that their origin might be linked with
rescattering Faisal (2009). In accordance with the results from circularly
polarised laser fields, the time-dependent Schrödinger equation (TDSE) results
also suggested that rescattering is involved in the creation of LES. Initial
theoretical investigations Yan  _et al._ (2010); Liu and Hatsagortsyan (2010)
attributed LES to be the result of interplay between forward scattering and
Coulomb focusing effect, which supported the proposition of recollision.
Another model suggested that the creation of LES is linked with the bunching
of photoelectrons, which miss the parent ion upon the so-called soft
recollisions Kästner  _et al._ (2012a, b). The insight into these low energy
features is of great importance for photoelectron holography to determine the
structural properties of the parent ion Meckel  _et al._ (2008); Bian and
Bandrauk (2012); Meckel  _et al._ (2014) and to explore the time-resolved
tunneling dynamics Porat  _et al._ (2018); Willenberg  _et al._ (2019).

In this work, we investigate the influence of relativistic nondipole effects
on the low energy features of the photoelectron momentum and energy spectra
obtained by using moderately intense linearly polarised near-infrared laser
field. These effects become discernible in the form of an offset in the
photoelectron momentum distribution along the laser propagation direction
Smeenk  _et al._ (2011a); Ludwig  _et al._ (2014); Ivanov  _et al._ (2016);
Haram  _et al._ (2019). The relativistic nondipole effects encompass the
short-range Coulomb focusing effect, demonstrating the breakdown of strong
field approximation and also of dipole approximation. In the former
approximation, the Coulomb interaction of the photoelectron with the parent
ion after the ionisation is neglected Marco (1992); Ivanov  _et al._ (2005).
The later approximation assumes that the electromagnetic field is spatially
uniform on the length scale of the electron motion during the laser pulse and
the influence of magnetic field component of the laser pulse on the electron
motion is negligible compared to that of the electric field component Reiss
(1992); Haram  _et al._ (2020). Breakdown of these approximations gives rise
to the relativistic nondipole effects which induce non-negligible momentum
transfer to the photoelectrons, which ultimately modifies the low energy
features of the photoelectron momentum and energy spectra.

We present high resolution, ultra-low energy photoelectron momentum and energy
spectra for the single ionisation of Ar by few-cycle, linearly polarised laser
pulses (800 nm, 6 fs, 0.8 PW/cm2) under exceptionally clean experimental
conditions. We check normalised asymmetry of the low energy features in the
laser propagation direction to study the impact of relativistic nondipole
effects on them. We show that the strong kinetic energy dependence of the
asymmetry is caused by the ionisation phase and thus by the trajectory of the
tunneled electron. For slow electrons, with their kinetic energy close to
zero, the propagation trajectory in the continuum allows these electrons to
experience strong Coulomb focusing. We show that such strong interaction can
cause a very interesting kinetic energy dependent peak shifting and spectral
narrowing on the transverse electron momentum distribution (TEMD). The
experimental results are supported by the truly relativistic ab initio three
dimensional time dependent Dirac equation (3D-TDDE) Haram  _et al._ (2019).

##  II Experimental System

The experiments were performed with a commercially available Ti:sapphire laser
system in combination with a reaction microscope (REMI). The laser system
delivers linearly polarised few cycle pulses in the near infrared (central
wavelength 800 nm) at 1 kHz repetition rate. The inherent ellipticity of the
laser pulses was eliminated by a quarter wave plate (QWP). The pulse duration
in the reaction zone was controlled by compensating the chirp using a pair of
fused silica wedges. The laser beam was tightly focused to a spot size of 7.25
$\mu$m in the reaction zone inside the ultrahigh vacuum chamber (10-10 mbar)
by a silver coated spherical mirror of focal length 75 mm. The laser intensity
was controlled by using pellicle beamsplitters and was chosen such that only
single ionisation of Ar remains dominant. The laser beam along y-axis was
crossed by the supersonic gas jet along x-axis at the laser focus from where
the resulting ions and electrons travel along z-axis/time of flight (TOF) axis
to their respective detectors of the REMI (see Fig. 1). A half-wave plate
(HWP) was used to rotate the polarisation axis of the laser pulses to TOF axis
of the REMI. The laser intensity in the reaction zone was determined precisely
by the recoil-ion momentum imaging method within 10% confidence interval
Alnaser  _et al._ (2004); Smeenk  _et al._ (2011b). To guarantee the
reliability of our data, laser intensity was constantly monitored during the
experiment with intensity fluctuations less than 5%.

The complete final state momentum and energy distributions of both electrons
and ions were recorded by REMI Ullrich  _et al._ (2003) with a controlled
well-defined momentum resolution. To avoid any obstruction in the electron
spectra, coincident detection of Ar+ ion allows to avoid any contributions of
electrons coming from higher ionisation states as we consider only those
electrons which conserve momentum with Ar+ ion. Moreover, the space charge
effects were avoided by carefully adjusting the laser intensity and width of
the supersonic gas jet in the reaction zone so that it does not influence the
final electron momentum distribution. A very low electric field (55 V/cm) was
applied parallel to the TOF axis and laser polarisation but perpendicular to
the laser beam and supersonic gas jet direction, which guides the low energy
electrons and ions towards two time and position sensitive micro-channel (MCP)
detectors. A homogeneous weak magnetic field (1.95 G) was applied to confine
the transverse motion of the electrons to ensure that the electrons with
energies less than 30 eV reach the detector. These low values of electric and
magnetic field improved the resolution of REMI such that we were able to
observe the low energy features with high resolution and clarity.

Figure 1:

The experimental setup used to explore the influence of relativistic nondipole
effects on the low energy features of momentum and kinetic energy spectra

Measurements were taken at two different laser intensities for comparison
purpose: 0.3 PW/cm2 and 0.8 PW/cm2. The value for the low intensity was chosen
such that the relativistic effects are least prominent (refer to our
previously published paper Haram  _et al._ (2019)) but can give us sufficient
count rate. The maximum laser intensity was selected so as to avoid
obstruction in the photoelectron momentum spectra as a result of the space
charge effect or multiple ionisation of the gas species.

##  III Theory

###  III.1 Theoretical model based on 3D-TDDE

To solve the TDDE we follow the procedure described in I.A.Ivanov (2015);
Ivanov (2017); Ivanov and Ho (2004); Haram  _et al._ (2019), which we briefly
recapitulate here. We solve the TDDE for the Ar atom in the field of a laser
pulse described by the vector potential $A_{z}(t-y/c)$ (the pulse is linearly
polarised in $z-$ direction and propagates in $y-$ direction):

$i{\frac{\partial\Psi({\bm{r}},t)}{\partial t}}={\hat{H}}\Psi({\bm{r}},t)\ ,$
(1)

where $\Psi({\bm{r}},t)$ is a four-component wave-function (bispinor),
${\hat{H}}$ the Hamiltonian operator:

$\hat{H}=\hat{H}_{\rm atom}+\hat{H}_{\rm int}\ ,$ (2)

where:

$\hat{H}_{\rm atom}=c{\bm{\alpha}}\cdot{\hat{\bm{p}}}+c^{2}(\beta-I)+IV(r),$
(3)

and

$\hat{H}_{\rm int}=c{\bm{\alpha}}\cdot{\bm{A}}(t,y)\ ,$ (4)

where $c=137.036\;a.u.$\- the speed of light in atomic units. We use the Dirac
basis for the $\bm{\alpha}-$ matrices:
$\displaystyle{\bm{\alpha}}=\left(\begin{array}[]{cc}{\bm{0}}&{\bm{\sigma}}\\\
{\bm{\sigma}}&{\bm{0}}\\\ \end{array}\right)$,
$\displaystyle\beta=\left(\begin{array}[]{cc}{\bm{I}}&{\bm{0}}\\\
{\bm{0}}&-{\bm{I}}\\\ \end{array}\right)$, $\displaystyle
I=\left(\begin{array}[]{cc}{\bm{I}}&{\bm{0}}\\\ {\bm{0}}&{\bm{I}}\\\
\end{array}\right)$, ${\bm{\sigma}}$ are Pauli matrices, ${\bm{0}}$ and
${\bm{I}}$ are $2\times 2$ null and identity matrices. We subtracted from the
field-free atomic Hamiltonian (3) the rest mass term $Ic^{2}$ so that we
operate in a more familiar energy scale commonly used in the non-relativistic
atomic physics calculations. Ar atom is described in the single active
electron (SAE) approximation, we use for $V(r)$ in Eq. (3) the model potential
given in Sarsa  _et al._ (2004).

Solution is represented as a series in basis bispinors:

$\Psi({\bm{r}},t)=\sum\limits^{J_{\rm max}}_{j\atop l=j\pm
1/2}\sum\limits_{M=-j}^{j}\Psi_{jlM}({\bm{r}},t),$ (5)

where each basis bispinor is:

$\Psi_{jlM}({\bm{r}},t)=\left(\begin{array}[]{c}g_{jlM}(r,t)\Omega_{jlM}({\bm{n}})\\\
f_{jlM}(r,t)\Omega_{jl^{\prime}M}({\bm{n}})\\\ \end{array}\right),$ (6)

and two-component spherical spinors are defined as
$\displaystyle\Omega_{jlM}({\bm{n}})=\left(\begin{array}[]{c}C^{jM}_{l\
M-{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}Y_{l,M-{\frac{1}{2}}}({\bm{n}})\\\
C^{jM}_{l\
M+{\frac{1}{2}}{\frac{1}{2}}-{\frac{1}{2}}}Y_{l,M+{\frac{1}{2}}}({\bm{n}})\end{array}\right)$,

(here $C^{jM}_{lm{\frac{1}{2}}\mu}$ are the Clebsch-Gordan coefficients,
$Y_{lm}({\bm{n}})$\- the spherical harmonics, and ${\bm{n}}={\bm{r}}/r$).
Parameters $l$ and $l^{\prime}$ in Eq. (5) must satisfy the relation
$l+l^{\prime}=2j$.

To take into account the non-dipole effects due to the spatial dependence of
the laser fields, vector potential in Eq. (4) is expanded in a series of
spherical harmonics at every step of the integration procedure. Substituting
expansion (5) in TDDE (1) and performing some cumbersome but straightforward
manipulations using well-known properties of spherical spinors Lifshitz and
Berestetskii (1982), one obtains a set of the coupled differential equations
for the radial functions $g_{jlM}(r,t)$ and $f_{jlM}(r,t)$ in Eq. (6). This
system is solved using a generalisation of the well-known matrix iteration
method (MIM) Nurhuda and Faisal (1999), which we described in detail in
I.A.Ivanov (2015).

Differential ionisation probabilities are calculated as
$P({\mu},{\bm{p}})=|a({\mu},{\bm{p}})|^{2}$, where $a({\mu},{\bm{p}})$ are the
ionisation amplitudes (here $\mu$ is polarisation (i.e., spin direction in the
electron’s rest frame), and ${\bm{p}}$ asymptotic electron momentum). The
amplitudes are obtained by projecting solution of the TDDE after the end of
the pulse on the set of the ingoing relativistic scattering states
$\Psi^{-}_{\mu,{\bm{p}}}({\bm{r}})$ of Ar atom calculated numerically using
the model potential $V(r)$ in Eq. (3). As we are not able to resolve different
electron spin states in the experiment, we sum our results for the probability
distributions over electron polarisation states.

##  IV Results and Discussion

The photoelectron momentum perpendicular to the laser polarisation plane, also
known as transverse momentum, carries the information about the recollision
event caused by Coulomb focusing of the photoelectrons Corkum (1993b); Brabec
_et al._ (1996); Blaga  _et al._ (2009); Quan  _et al._ (2009b); Korneev  _et
al._ (2012a); Liu  _et al._ (2012) without being disturbed by the momentum
transfer from the laser field. Therefore, TEMD is considered as the most
suitable spectrum to explore the influence of relativistic nondipole effects
on the low energy features. In addition, the dependence of transverse momentum
on the kinetic energy of photoelectrons provides more insight into the
underlying physics.

The photoelectron momentum distribution recorded in our experiment from the
strong field ionisation of Ar at intensity 0.8 PW/cm2 has sufficiently high
resolution to resolve the low energy features. They are revealed as a
superposition of multiple symmetric patterns about the polarisation axis. In
Fig. 2, we can see that the higher energy region of the spectrum contains a
series of intercycle features in the form of ATI rings, whereas intracycle
fan-like stripes emerging radially outwards in the range
$-0.3\><p_{\parallel}<\>0.3$ can be seen in the low energy regions. The low
energy features do not exactly match in Fig. 2 (a) and (b), as the intensity
could be slightly different in experiment and simulation, and volume effects
which are not taken into account in the simulation.

Figure 2:

Longitudinal (p||=pz) vs transverse (p⟂=$\sqrt{p_{x}^{2}+p_{y}^{2}}$) electron
momentum distribution resulting from the strong field ionisation of Ar using
few cycle pulses at 0.8 PW/cm2, representing the intercycle and intracycle
interference features. (a) theoretical simulation (b) measurement results.

###  IV.1 Momentum/Kinetic energy dependence

The influence of relativistic nondipole effects on the low energy
photoelectrons can be seen in the plot of transverse momentum $P_{y}$ along
the laser propagation direction vs photoelectron energy, showing the tilting
for very low energy photoelectrons (0–0.5 eV). This tilt is absent for the
transverse momentum $P_{x}$ perpendicular to the laser propagation direction
(see Fig. 3).

Figure 3:

Energy resolved transverse electron momentum spectrum at 0.8 PW/cm2. (a) low
energy electrons for transverse momentum along the laser propagation direction
experience a tilt in the negative direction and then in the positive direction
with increasing kinetic energy. (b) no tilting in the low energy region for
the transverse momentum perpendicular to the laser propagation direction, but
a spread can be observed at higher kinetic energy.

The photoelectrons experience strong Coulomb focusing effects due to the
rescattering electron trajectories that correspond to very low energy. As the
low energy photoelectrons typically get ionised at the peak of the laser field
and return to the parent ion with close to zero momentum, therefore they stay
longer around the ion and experience stronger Coulomb attraction. The
interaction between the Coloumb focusing and relativistic nondipole effects
leads to the spectral narrowing of the low energy TEMD with the peak shift in
the negative direction. However, the high-energy TEMD gets broadened with a
forward peak shift. This energy-dependent offset can be used as a self-
referencing technique for the detection and analysis of the relativistic
nondipole effect, without relying on the auto-ionisation of Rydberg state
molecules to calibrate the transverse momentum offset Smeenk  _et al._
(2011a); Ludwig  _et al._ (2014).

At the intensity 0.8 PW/cm2, it was already expected and has been already
reported that the photoelectrons show a negative peak shift. As can be seen in
the results from our previous work (Fig. 4), which compares the experimental
data obtained in the intensity range 0.6–3.0 PW/cm2 with the 3D-TDDE
simulation results in the intensity range 0.5–7.0 PW/cm2, the peak
corresponding to the very low energy electrons continues shifting in the
negative direction with increasing intensity Haram  _et al._ (2019).

Figure 4:

With increasing intensity, the relative peak shift of the TEMD increases in
the negative direction.

The comparison of asymmetry as observed in the simulated TEMD along the laser
propagation direction at two different intensities 0.3 PW/cm2 and 0.8 PW/cm2
is presented in Fig. 5. It is evident from the figure that the momentum
distribution is symmetric about zero momentum at low intensity Fig. 5(a).
However, at a higher intensity, not only the very low energy electrons are
shifted in the negative direction, but also the tilt of the photoelectron
momentum distribution is quite obvious Fig. 5(b).

Figure 5:

Simulation results: Normalised transverse vs longitudinal momentum spectra at
(a) low (0.3 PW/cm2) and (b) high (0.8 PW/cm2) intensities.

The quantitative analysis of this tilt is performed by determining the peak
position of the transverse momenta $P_{y}$ along laser propagation direction,
integrated over the energy range 0.2 eV as a function of photoelectron energy
by employing the same procedure as used in Haram  _et al._ (2019). As shown in
the Fig. 6, the zero energy or close to zero energy electrons show a negative
peak shift i.e. -3 x 10-3 a.u., which keeps on increasing in the negative
direction for the photoelectrons having energy less than 0.3 eV. The cause of
this negative shift is the strong Coulomb focusing effect for electrons having
close to zero energy. However, the very low energy photoelectrons in the range
0.3–1.3 eV experience a push in the forward direction because of the radiation
pressure, causing a net peak shift of 6 x 10-3 a.u.. With increasing energy,
instead of continuing forward shift the photoelectrons experience a negative
peak shift again. The shift pattern continues to oscillate from
negative–positive–negative shift for the photoelectron having energy greater
than 1.9 eV but in a smaller interval of approximately 1 x 10-3 a.u.. The
overall

Figure 6:

The overall $P_{y}$ resolved peak shift is in the forward direction: 7 x 10-3
a.u. for 0.3 eV electrons increasing to 1 x 10-2 a.u. for 10 eV electrons.
However, for very low energy electrons the peak shift is in the negative
direction, which reverses its direction for the electrons greater than 0.3 eV
and then follows oscillatory behaviour with increasing energy after 1.9 eV.

peak shift, however, keeps on increasing from 7 x 10-3 a.u. to 1 x 10-2 a.u..

The same trend of the peak shift has been observed for the simulation results
obtained at even higher intensity. Fig. 7 compares the experimental data
obtained at 0.8 PW/cm2 with the simulation results obtained at 1.4 PW/cm2,
which reflects that the overall shift might be different as a function of
intensity. However, close to zero energy and the very low energy electrons,
the main reason of negative peak shift is due to Coulomb focusing and hence
they behave the same way in this intensity range.

Figure 7:  Energy resolved transverse electron momentum spectra: (a)
simulation results at 1.4 PW/ cm2, (b) experimental results at 0.8 PW/ cm2.

To further explore the tilt caused by the relativistic nondipole effects on
the low energy photoelectrons in Fig. 3, we plot transverse momentum in the
laser propagation direction vs longitudinal momentum. Fig 8 presents the
comparison between the simulated

Figure 8:

Normalised transverse vs longitudinal momentum spectra at high intensity 0.8
PW/ cm2. Comparison of (a) simulation and (b) experimental results. (c) The
plot of extracted peak shift from the experimental data.

spectrum and measured TEMD at 0.8 PW/cm2. An overall tilt in Py in both (a)
simulated spectra and (b) measured spectra is observed. To highlight the
features of interest, we zoom in the transverse momentum in a narrow range
[-0.05 a.u., 0.05 a.u.]. and extract the peak position of the transverse
momentum P⟂ integrated over the momentum range 0.02 a.u.. as shown in the (c)
plot. The peak shift is extracted by the same procedure as described in Haram
_et al._ (2019). The peak shift is determined by the asymmetry in the cusp of
the TEMD. For this purpose, the function $V(P_{\perp})\equiv\ln{W(P_{\perp})}$
is analysed in a narrow range of transverse momenta $|P_{\perp}|\leq 0.1$
a.u., where $W(P_{\perp})$ is the ionisation rate. The experimental data is
fitted with the function $V(P_{\perp})=B+A|P_{y}-\beta|^{\alpha}$ by
performing a series of least square fits, where $A$, $B$, $\alpha$ and $\beta$
are the fitting parameters with $\beta$ accountable for the peak shift
$\left\langle P_{\perp}\right\rangle$. The same procedure has been employed to
determine the $P_{y}$ resolved peak shift in Fig. 6.

For zero and close-to-zero energy electrons the peak shift is negative, which
keeps on oscillating between negative and positive direction with increasing
longitudinal momentum but shows an overall positive shift i.e. a shift from -
3 x 10-3 a.u. to 4 x 10-3 a.u.. This oscillatory behavior of the peak shift
can be attributed to the photoelectrons with different trajectories.

Further investigation on the TEMD along the laser propagation direction for
the low energy electrons (0-1 eV) integrated over the momentum range 0.1 a.u.
obtained at 0.8 PW/ cm2 shows quite interesting behaviour. For close to zero
energy electrons (0-0.01 eV) , the TEMD seems to be symmetric and uniform.
However, the wings of the TEMD start to appear progressively with increasing
energy in the range 0.15-0.4 eV, which gradually fade away leading to a
broader distribution after this energy range. The central peak of the TEMD
shifts in the negative direction for the electrons in the energy range
0.16-0.25 eV as shown in Fig 9 (a). These are the low energy electrons with
long trajectories that may revisit

Figure 9:

Measured TEMD along the laser propagation direction obtained 0.8 PW/cm2. (a)
Wings of TEMD start to appear for the electrons in the energy range 0.16-0.25
eV with an asymmetric distribution (blue) compared to the distribution for
very low energy electrons (grey). (b) More pronounced wings of the TEMD for
the electrons in the energy range 0.25-0.36 eV. The central peak is shifting
in the negative direction whereas wings tend to shift in the positive
direction.

the parent ion Chelkowski  _et al._ (2015). This peak shift keeps on
increasing in the negative direction until the electrons gain energy of $\sim$
0.36 eV. In contrast, the wings of the electrons are shifted in the positive
direction, since they are formed by the direct electrons which never get a
chance revisit the parent ion (see Fig 9 (b)). The high energy electrons with
energy greater than 0.36 eV corresponding to short trajectories contribute to
the overall positive peak shift.

###  IV.2 CEP dependence

CEP averaged momentum distributions

Figure 10:

CEP averaged photoelectron momentum distributions. (a) at low intensity 0.3
PW/cm2, (b) at high intensity 0.8 PW/cm2.

The simulated CEP averaged photoelectron momentum distributions based on
3D-TDDE at 0.3 PW/cm2 and 0.8 PW/cm2 are as shown in Fig. 10 (a,b). In
general, these spectra exhibit a number of features, mainly consisting of
intercycle and intracycle interferences. Intercycle interferences cause the
momentum distribution to evolve into concentric ATI-rings centered at zero
momentum and are formed as a result of superposition of electron trajectories
corresponding to complex release time during multiple optical cycles. These
rings are insensitive to the Coulomb potential and their yield depends upon
the direction of photoelectron emission Arbó  _et al._ (2006, 2012). Regular
carpet-like features are created by the interference of ATI rings that are
separated by a number of cycles Korneev  _et al._ (2012b). Intracycle
interferences lead to the formation of fan-like stripes caused by the
interference of electron trajectories within one optical cycle. These include
the direct photoelectron trajectories and the trajectories corresponding to
those electrons which are deflected by the Coulomb potential without
experiencing any hard recollisions Lai  _et al._ (2015). The intracycle
interferences are also responsible for the coherent superposition of distinct
states known as Freeman resonances. These resonant features are formed as a
result of a range of intensities attributed to the focal volume below the peak
intensity Freeman  _et al._ (1987). The pattern of these fan-like features
exhibit strong dependence on the wavelength since it influences the states to
be populated with a particular angular quantum number ($l$) Marchenko  _et
al._ (2010). The fine holographic features beside the main features emerge
from the intracycle interferences between the direct photoelectron
trajectories and the trajectories that correspond to hard recollisions Maxwell
_et al._ (2017).

It has been established that the features in the photoelectron momentum
distributions are sensitive to the laser wavelength and intensity Marchenko
_et al._ (2010, 2011). Since, we are comparing the momentum distributions at
two quite different intensities, the number of ATI-rings and fan-like stripes
as well as their patterns are found to be significantly different (see Fig.
10). Considering these momentum distributions are CEP averaged, the features
are found to be symmetric about zero momentum.

CEP dependent momentum distributions

The CEP has profound effect on the photoelectron momentum distributions. The
spectra obtained at two different intensities show quite unique features for
each CEP as shown in Fig. 11 and Fig. 12.

Figure 11:

CEP dependent photoelectron momentum distributions at low intensity 0.3
PW/cm2.

Figure 12:

CEP dependent photoelectron momentum distributions at high intensity 0.8
PW/cm2.

The features are significantly different for different CEP values i.e. not
alike nor symmetric, since a specific CEP value may enhance the population of
only a particular resonant feature. However, as reported earlier in our
previous paper, the effect of CEP on the nondipole effects is not resolvable
within experimental uncertainty. In order to explore these effects, the peak
shift of the TEMD along the laser propagation direction was extracted for the
sine and cosine pulses, which turned out to be the much smaller than the
experimental uncertainty Haram  _et al._ (2019). Further investigation based
on the simulation results at 0.3 PW/cm2 and 0.8 PW/cm2 supports the previous
results showing that for different CEP values the peak shift is extremely
small especially at lower intensity (see Fig. 13).

Figure 13:

CEP dependent peak shift of the TEMD along laser propagation direction at 0.3
PW/cm2 and 0.8 PW/cm2.

##  V Classical Analysis

To investigate the role of photoelectron trajectories on the relativistic
nondipole effects, a simple physical picture based on a modified three step
model is presented. In the first step, the electron is ionised close to the
peak of laser electric field. In the next step, the photoelectron is
accelerated by the driving laser field along its polarisation axis and follows
its oscillatory motion with a constant drift motion given by the laser field
at the tunnel exit. Due to this drift momentum, which the photoelectron
possesses when the laser pulse is over, the photoelectron gets detected at the
detector. However, perpendicular to the polarisation axis, the photoelectron
is pushed along its propagation axis with a constant velocity due to the
Lorentz force. In the third step, the photoelectron obtains a negative
transverse momentum when it revisits and gets elastically scattered by the
Coulomb potential of parent ion, which is similar to the Rutherford scattering
process. The drift momentum gained in the second step is responsible for the
particular trajectories that allows the low energy rescattering and cause the
photoelectron to be strongly focused. Thus, the negative photoelectron
momentum shift is sensitive to the photoelectron trajectory, and it happens
within one optical cycle.

Classical electron trajectory model:

The validity of this modified three step model can be confirmed by performing
a pure classical electron trajectory simulation for different ionisation
times. Experimentally, it can be confirmed by using CEP locked few-cycle pulse
or two-color laser field to break the top/bottom symmetry, which is the
subject of our next work. In this way, the photoelectron trajectory and its
momentum shift along laser propagation direction are expected to be different
for P${}_{||}>0$ and P${}_{||}<0$.

Here, we have performed a full classical electron trajectory simulation. The
pump laser is propagating along y-axis and its electric field component is
linearly polarised along z-axis:

$\begin{pmatrix}E_{x}\\\ E_{y}\\\ E_{z}\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\
E_{0}\;sin^{2}(\frac{\pi(t-\frac{y}{c})}{\tau_{0}})\;cos[\omega(t-\frac{y}{c})+\phi_{0}]\end{pmatrix}$
(7)

where $E_{0}$ is the peak electric field strength (0.06 a.u.), $\tau_{0}$ is
the pulse duration (5 optical cycle), $\omega$ is the pump laser carrier
frequency, $\phi_{0}$ is the carrier-to-envelope phase (CEP), and c is the
speed of light in vacuum. The magnetic field component of the pump laser is
linearly polarised along x-axis:

$\begin{pmatrix}B_{x}\\\ B_{y}\\\ B_{z}\end{pmatrix}=\begin{pmatrix}-E_{z}\\\
0\\\ 0\end{pmatrix}$ (8)

The dynamics of electron in the presence of pump electric and magnetic field,
as well as the Coulomb potential force from the ion is given by:

$\begin{pmatrix}F_{x}\\\ F_{y}\\\
F_{z}\end{pmatrix}=\begin{pmatrix}-\frac{dU(r)}{dr}\frac{x}{r}\\\
\frac{B_{x}}{c}v_{z}-\frac{dU(r)}{dr}\frac{y}{r}\\\
-eE_{z}-\frac{B_{x}}{c}v_{y}-\frac{dU(r)}{dr}\frac{z}{r}\end{pmatrix}$ (9)

where $U(r)=-1/\sqrt{r^{2}+s}$ is a softcore Coulomb potential with
$s=0.1\;a.u.$ , $r=\sqrt{x^{2}+y^{2}+z^{2}}$ is the distance between electron
and ion.

Figure 14: (a) The calculated typical electron trajectory with negative
momentum along laser propagation direction, where the initial exit position
$z_{0}=6.97\;a.u.$, initial velocity $vz_{0}=-0.184\;a.u.$ and
$t_{0}=2.57\;o.p.$ is used in the classical simulation. (b) The time evolution
of the driving electric field (blue dashed curve) and time dependent electron
momentum $P_{y}$. The point A, B, C in (a) indicates the electron position
when $t=2.57\;o.p.$ (point A in (b)), $t=2.81\;o.p.$ (point B in (b)), and
$t=2.996\;o.p.$ (point C in (b)) respectively.

$v_{x},v_{y},v_{z}$ are the three electron velocity components. Given initial
conditions, including electron tunneling time $t_{0}$, exit site $(0,0,z_{0})$
and assuming zero initial velocity after tunneling ionisation, the final 3D
momentum and position of the free electron can be calculated by solving the
Newton function (9), where an integration time of 7 optical periods (o.p.) is
used. By scanning the tunneling time $t_{0}$ and exit site position $z_{0}$,
we found the negative peak shift of the TEMD that corresponds to the electrons
having tunneling time at around the local peaks of the driving electric field
and the exit site around $z_{0}=6.97\;a.u.$.

Fig. 14 shows a typical electron trajectory with negative momentum shift along
y-axis, where the electron is ionized at $t_{0}=2.57\;o.p.$ (electric field is
peaked at 2.5 o.p.), the exit site is $(x_{0}=0,y_{0}=0,z_{0}=6.97\;a.u.)$,
and exit velocity is $(vx_{0}=0,vy_{0}=0,vz_{0}=-0.184\;a.u.)$ Within half
optical cycle, the electron is driven away from the parent ion and reaches
largest ion/electron distance at $t=2.81\;o.p.$. After that, the electron is
driven back towards the ion, and gains a positive spatial displacement along
laser propagation direction (y-axis) due to the radiation pressure produced by
the laser field. At t = 3.0, the electron is re-scattered by the parent ion
and gains a negative momentum which is opposite to the laser propagation
direction. An electron trajectory with positive momentum along y-axis is
possible for electrons which have not experienced re-scattering. Figure 15
shows a typical electron trajectory with positive momentum shift the electron
is no longer driven back to its parent ion, the electron can gain positive
momentum along y-axis.

Figure 15:

Same as Fig. 2, but with ionisation time set as $t_{0}=2.43\;o.p.$, the exit
site as $(x_{0}=0,y_{0}=0,z_{0}=6.97\;a.u.)$, and exit velocity as
$(vx_{0}=0,vy_{0}=0,vz_{0}=0.184\;a.u.)$. The point A, B, C in (a) indicates
the electron position when $t=2.43\;o.p.$ (point A in (b)), $t=2.81\;o.p.$
(point B in (b)), and $t=4.9\;o.p.$ (point C in (b)) respectively.

##  VI Conclusion and Outlook

We have investigated how the low-energy features of the photoelectron momentum
and energy spectra are impacted by the relativistic nondipole effects for
ultrashort linearly polarised near-infrared laser field. The measured and
simulated photoelectron momentum spectra for Ar show very interesting sharp
features in the form of ATI-rings and Freeman resonances due to the inter- and
intra-cycle resonances. Further investigations on the energy-resolved
relativistic nondipole effects reveal an asymmetry or tilt in both momentum
and energy spectra. It shows that the central peak shift for the low-energy
electrons is always in the negative direction with reference to the laser
propagation direction. However, the high energy electrons that also form the
wings of the TEMD contribute towards the positive shift. Moreover, this peak
shift is found to be oscillating between negative and positive direction,
which can be ascribed to specific electron trajectories as explored by
classical analysis based on modified three step model. The influence of CEP on
the relativistic nondipole effects has also been investigated, showing that it
has a very small effect on the peak shift of the TEMD such that it is
difficult to resolve it within experimental uncertainty.

## Acknowledgement

This research was supported by the Australian Research Council Discovery
Project DP110101894. We also acknowledge support from the Institute for Basic
Science, Gwangju, Republic of Korea, under IBS-R012-D1. N.H., A.A., U.S.S. and
D.C. are supported by Griffith University International Postgraduate Research
Scholarship (GUIPRS). H.X. is supported by an ARC Discovery Early Career
Researcher Award DE130101628.

N.H. and H.X. contributed equally to this work.

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# Some Ramsey-type results

Jin<EMAIL_ADDRESS>
School of Mathematical Sciences
University of Science and Technology of China, Hefei, Anhui 230026, China.

###### Abstract

The Ramsey’s theorem says that a graph with sufficiently many vertices
contains a clique or independent set with many vertices. Now we attach some
parameter to every vertex, such as degree. Consider the case a graph with
sufficiently many vertices of large degree, we can get a similar Ramsey-type
result. The Ramsey’s theorem of connected version says that every connected
graph with sufficiently many vertices contains an induced path, clique or star
with many vertices. Now we require the vertex is non-trivial, i.e. the
parameter of this vertex is non-trivial, such as $\operatorname{deg}(v)\geq
2$. A connected graph with sufficiently many non-trivial vertices must contain
some special induced subgraph. We also get the non-connected version of this
Ramsey-type result as a corollary.

Keywords: Ramsey theory, graph parameter, $h$-index.

##  1 Introduction

For positive integers $m$ and $n$ with $m<n$, let $[m,n]$ denote the set
$\\{m,m+1,\dots,n-1,n\\}$, and let $[n]$ denote $[1,n]$. In this paper, we
consider only finite, undirected and simple graphs. Let $G$ be a graph. For
vertices $y,z\in V(G)$, we use the symbol $y\sim z$ to mean that $y$ and $z$
are adjacent. $\mathbbm{1}_{y\sim z}$ is the characteristic function which
equals $1$ if $y$ is adjacent to $z$, $0$ otherwise. For a vertex $v\in V(G)$,
let $N(v)$ denote the neighborhood of $v$ in $G$, thus $N(v)=\\{u\in
V(G):u\sim v\\}$. For a subset $S\subset V(G)$, we denote by $G[S]$ the
subgraph of $G$ induced by $S$, i.e. $V(G[S])=S,\,E(G[S])=\\{\\{u,v\\}:u,v\in
S,u\sim v\mbox{ in}\;G\\}$. Let $\alpha(S)$ be the independence number of the
induced subgraph $G[S]$, $c(S)$ be the number of connected components of
$G[S]$. We say $v$ is complete to $S$ if $S\subset N(v)$. For two disjoint
subsets $A,B$ of $V(G)$, we denote by $G[A,B]$ the bipartite subgraph (not
induced if $E(G[A])\cup E(G[B])\neq\emptyset$.) with two parts $A$ and $B$
formed by the edges of $G$ between $A$ and $B$, i.e. $V(G[A,B])=A\sqcup B$,
$E(G[A,B])=\\{\\{a,b\\}:a\in A,\,b\in B,\,a\sim b\mbox{ in }G\\}$. Stable set
has the same meaning as independent set. The more notations, see [5].

For graphs $H_{1},H_{2}$, we say $H_{1}\leq H_{2}$ if $H_{1}$ is an induced
subgraph of $H_{2}$. For a family $\mathcal{H}$ of graphs, we say $G$ is
$\mathcal{H}$-free, if there is no graph $H\in\mathcal{H}$ such that $H\leq
G$. For families $\mathcal{H}_{1},\,\mathcal{H}_{2}$, we say
$\mathcal{H}_{1}\leq\mathcal{H}_{2}$ if any graph in $\mathcal{H}_{2}$ is not
$\mathcal{H}_{1}$-free, i.e.
$\forall\,H_{2}\in\mathcal{H}_{2},\,\exists\,H_{1}\in\mathcal{H}_{1}$ such
that $H_{1}\leq H_{2}$. Specially, $H_{1}\leq H_{2}$ is equivalent to
$\\{H_{1}\\}\leq\\{H_{2}\\}$. Note that if
$\mathcal{H}_{1}\leq\mathcal{H}_{2}$, then every $\mathcal{H}_{1}$-free graph
is also $\mathcal{H}_{2}$-free. This language was introduced in [7] firstly.

In 1929, the well-known Ramsey’s theorem appeared.

######  Theorem 1.1.

[11] For any positive integers $m,n$, there exists a minimum positive integer
$R=R_{m}(n)$ such that if the edges of the complete graph $K_{R}$ are colored
with $m$ colors, then there is a monochromatic clique of order $n$, i.e. a
clique all of whose edges have the same color.

Specially, for $m=2$, Ramsey’s theorem says that for any large $n$, a graph
with $R_{2}(n)$ vertices contains $K_{n}$ or $E_{n}$ as an induced subgraph.
$E_{n}$ is not connected. When we restrict our attention to connected graphs,
we can get connected induced subgraph better than $E_{n}$.

######  Proposition 1.2.

[5] For any positive integer $n$, there exists an integer $N_{0}(n)$ such that
every connected graph on at least $N_{0}(n)$ vertices contains $P_{n},K_{n}$
or $K_{1,n}$ as an induced subgraph.

Lozin [10] introduces a general philosophy of Ramsey-type problems: given a
graph parameter $\mu$, characterize the forbidden graphs condition to bound
the parameter $\mu$, i.e. find all graph families $\mathcal{H}$ such that
there is a constant $c$ satisfying $\mu(G)\leq c$ for any connected
$\mathcal{H}$-free graph $G$. When $\mu$ is the order of graph $G$, this
philosophy leads to the classical Ramsey’s theorem. Atminas, Lozin, and Razgon
[2] studied the parameter $\mu$ as the length of longest path in graph $G$.
Chiba and Furuya [3] studied the parameter path cover number. Furuya [8]
studied the parameter domination number. Many other parameters have been
studied in [1, 4, 10].

Now we consider the parameter $\mu$ as the variation of the order of graph
$G$. Compared to Proposition 1.2, we restrict that the graph has many
nontrivial vertices to get better induced subgraphs. Here we attach some
parameter $p$ to every vertex and vertex $v$ is nontrivial if $p(v)\geq 2$.
Specifically, $p(v)$ will be $\deg(v)$, $\alpha(N(v))$, $c(N(v))$, and the
adhesion of $v$. We define the adhesion of $v$ as $c(G-v)-c(G)+1$, and denote
it by $\operatorname{adh}(v)$. Thus $v$ is a cut vertex if and only if
$\operatorname{adh}(v)\geq 2$. Therefore the adhesion of a vertex is a
generalization of cut vertex. Notice that for a vertex $v\in V(G)$ , we have
$\operatorname{deg}(v)\geq\alpha(N(v))\geq c(N(v))\geq\operatorname{adh}(v)$.

In order to state our Ramsey-type results, we introduce some kinds of graphs.
As usual, we let $K_{n},E_{n},P_{n}$ and $K_{s,t}$ denote the complete graph,
edgeless graph, path and complete bipartite graph respectively. For two graphs
$G_{1}$ and $G_{2}$, we define the join $G_{1}+G_{2}$ by the graph with vertex
set $V(G_{1})\cup V(G_{2})$ and edge set $E(G_{1})\cup
E(G_{2})\cup\\{\\{x,y\\}:x\in V(G_{1}),\,y\in V(G_{2})\\}$. We denote by $nG$
the graph consisting of $n$ disjoint copies of $G$. The more kinds of graphs,
see Figure 1.

  * •

$K_{1,n}^{*}$: The graph obtained by adding a pendant to every leaf of
$K_{1,n}$.

  * •

$K_{n}^{*}$: The graph obtained by adding a pendant to every vertex of
$K_{n}$.

  * •

$CK_{n}$: The graph obtained by adding a perfect matching between two disjoint
copies of $K_{n}$.

  * •

$T_{n}$: The graph obtained by adding a new vertex adjacent to all vertices of
$E_{n}$ from $K_{n}+E_{n}$.

  * •

$K_{n}^{n}$: The graph obtained by adding $n$ pendents to every vertex of
$K_{n}$.

$\vdots$$K_{1,n}^{*}$$\vdots$$K_{n}$$K_{n}^{*}$$\vdots$$K_{n}$$CK_{n}$$\vdots$$K_{n}$$T_{n}$$K_{n}$$\cdots$$\vdots$$\vdots$$\cdots$$K_{n}^{n}$
Figure 1: The graphs $K_{1,n}^{*},\,K_{n}^{*},\,CK_{n},\,T_{n}$ and
$K_{n}^{n}.$

Requiring the graph has many nontrivial vertices, we have the following
related Ramsey-type results.

######  Theorem 1.3.

There is a constant $c=c(\mathcal{H})$ such that

$\\#\\{v\in V(G):\deg(v)\geq 2\\}\leq c$

for every connected $\mathcal{H}$-free graph $G$ if and only if

$\mathcal{H}\leq\\{K_{n},P_{n},K_{1,n}^{*},K_{2,n},K_{2}+nK_{1},K_{1}+nK_{2}\\}$

for some positive integer $n$.

######  Theorem 1.4.

There is a constant $c=c(\mathcal{H})$ such that

$\\#\\{v\in V(G):\alpha(N(v))\geq 2\\}\leq c$

for every connected $\mathcal{H}$-free graph $G$ if and only if

$\mathcal{H}\leq\\{K_{n}^{*},P_{n},K_{1,n}^{*},K_{2,n},E_{2}+K_{n},K_{1}+nP_{3},CK_{n}\\}$

for some positive integer $n$.

######  Theorem 1.5.

There is a constant $c=c(\mathcal{H})$ such that

$\\#\\{v\in V(G):c(N(v))\geq 2\\}\leq c$

for every connected $\mathcal{H}$-free graph $G$ if and only if

$\mathcal{H}\leq\\{K_{n}^{*},P_{n},K_{1,n}^{*},K_{2,n},CK_{n},T_{n}\\}$

for some positive integer $n$.

######  Theorem 1.6.

There is a constant $c=c(\mathcal{H})$ such that

$\\#\\{v\in V(G):\operatorname{adh}(v)\geq 2\\}\leq c$

for every connected $\mathcal{H}$-free graph $G$ if and only if

$\mathcal{H}\leq\\{K_{n}^{*},K_{1,n}^{*},P_{n}\\}$

for some positive integer $n$.

To be more precisely, we give another example. Using Ramsey’s theorem easily,
we have the following.

######  Proposition 1.7.

There is a constant $c=c(\mathcal{H})$ such that $\Delta(G)\leq c$ for every
$\mathcal{H}$-free graph $G$ if and only if
$\mathcal{H}\leq\\{K_{n},K_{1,n}\\}$ for some positive integer $n$.

We know that for any positive integer $n$, a graph with sufficiently large
maximum degree has sufficiently many vertices and hence contains $K_{n}$ or
$E_{n}$ as an induced subgraph. However, $\Delta(E_{n})=0$ is not large. Thus
for family $\mathcal{H}$ of graphs, if $\mathcal{H}$ satisfies the condition
that there is a constant $c=c(\mathcal{H})$ such that $\Delta(G)\leq c$ for
every $\mathcal{H}$-free graph $G$, we can not get the conclusion that
$\mathcal{H}\leq\\{K_{n},E_{n}\\}$. The right conclusion is
$\mathcal{H}\leq\\{K_{n},K_{1,n}\\}$. Here $K_{1,n}$ is a better induced
subgraph than $E_{n}$. This example gives an explanation what the better
means.

Let us continue to look at this example. Observe that for the star $K_{1,n}$,
it has only one vertex with large degree, and other vertices are degree $1$.
In some sense, it is unfair for most vertices to say $K_{1,n}$ has large
maximum degree. Now we consider the case where there are many vertices of
large degree.

######  Theorem 1.8.

There are two constants $c_{1}=c_{1}(\mathcal{H}),c_{2}=c_{2}(\mathcal{H})$
such that

$\\#\\{v\in V(G):\deg(v)\geq c_{1}\\}<c_{2}$

for every $\mathcal{H}$-free graph $G$, if and only if
$\mathcal{H}\leq\\{K_{n},K_{n,n},nK_{1,n}\\}$ for some positive integer $n$.

Note that [1] got the same result as Theorem 1.8 in the language of minimal
hereditary classes. For a parameter $p$ of $V(G)$, the $h$-index of $G$ is the
largest positive integer $k$ such that $G$ has $k$ vertices of parameter $p$
at least $k$. It’s introduced in [9] and important in the study of dynamic
algorithms [6]. When the parameter $p$ is vertex degree, Theorem 1.8
characterizes the family $\mathcal{H}$ such that there is a uniform bound of
$h$-index for any $\mathcal{H}$-free graph $G$. Similarly, we replace degree
by other parameters of a vertex, then get the following.

######  Theorem 1.9.

The following are equivalent:

(1) There are two constants
$c_{1}=c_{1}(\mathcal{H}),c_{2}=c_{2}(\mathcal{H})$ such that

$\\#\\{v\in V(G):\alpha(N(v))\geq c_{1}\\}<c_{2}$

for every $\mathcal{H}$-free graph $G$.

(2) There are two constants
$c_{1}=c_{1}(\mathcal{H}),c_{2}=c_{2}(\mathcal{H})$ such that

$\\#\\{v\in V(G):c(N(v))\geq c_{1}\\}<c_{2}$

for every $\mathcal{H}$-free graph $G$.

(3) $\mathcal{H}\leq\\{K_{n,n},nK_{1,n},K_{n}+E_{n},K^{n}_{n}\\}$ for some
positive integer $n$.

######  Theorem 1.10.

There are two constants $c_{1}=c_{1}(\mathcal{H}),c_{2}=c_{2}(\mathcal{H})$
such that

$\\#\\{v\in V(G):adh(v)\geq c_{1}\\}<c_{2}$

for every $\mathcal{H}$-free graph $G$, if and only if
$\mathcal{H}\leq\\{nK_{1,n},K^{n}_{n}\\}$ for some positive integer $n$.

In section $2$, we will consider the case a connected graph $G$ has many
nontrivial vertices, and prove Theorem 1.3, Theorem 1.4, Theorem 1.5 and
Theorem 1.6. In section $3$, we will consider the case a graph $G$ has many
vertices of large parameter, and prove Theorem 1.8, Theorem 1.9 and Theorem
1.10. In section $4$, we will consider the case a graph has many nontrivial
vertices, and give the corresponding corollaries. We will also give some
remarks of our results.

##  2 Connected graph with many nontrivial vertices

The following lemma was implicitly proved and used in [3] and [8]. Actually,
it can date back to earlier, the definition of irredundant set. Here we put it
in the spotlight.

######  Lemma 2.1.

For a bipartite graph $G=(X,Y,E)$, the minimum degree of vertices in $X$
satisfies $\delta(X)\geq 1$, the maximum degree of vertices in $Y$ satisfies
$\Delta(Y)\leq n$. The induced matching number of $G$ satisfies
$\mu^{\prime}(G)\geq p$ provided that $|X|\geq n(p-1)+1$, and the bound is
tight.

###### Proof.

Since $\delta(X)\geq 1$, $Y$ dominates $X$, i.e. $X\subset N(Y)$. Take minimal
subset $Y^{\prime}$ of $Y$ dominates $X$, i.e. $N(Y^{\prime})=X$ and for any
proper subset $Y^{\prime\prime}$ of $Y^{\prime}$, $N(Y^{\prime\prime})\not=X$.
So $Y^{\prime}$ is irredundant, i.e. every $y\in Y^{\prime}$ has a private
neighbor $x_{y}\in X$ such that $N(x_{y})\cap Y^{\prime}=\\{y\\}$. Since
$\Delta(Y)\leq n$, every vertex can dominate at most $n$ vertices in $X$.
$|X|\geq n(p-1)+1$, thus $|Y^{\prime}|\geq p$. Choose $p$ vertices in
$Y^{\prime}$, and a private neighbor for everyone, they form an induced
$pK_{2}$. So $\mu^{\prime}(G)\geq p$.

Specially, let $G=(X,Y,E)$ be $X=\\{x_{ij}:i\in[p-1],\,j\in[n]\\}$,
$Y=\\{y_{i}:i\in[p-1]\\}$, $E=\\{(x_{ij},y_{i}):i\in[p-1],\,j\in[n]\\}$. It
satisfies $\delta(X)\geq 1$, $\Delta(Y)\leq n$, but $\mu^{\prime}(G)=p-1$
since $|X|$ only equals to $n(p-1)$. ∎

###### Proof of Theorem 1.3.

We first prove the “only if” part. Suppose $\mathcal{H}$ satisfies there is a
constant $c=c(\mathcal{H})$ such that $\\#\\{v\in V(G):\deg(v)\geq 2\\}\leq c$
for every connected $\mathcal{H}$-free graph $G$. The number of vertices with
degree at least $2$ for

$K_{n},P_{n},K_{1,n}^{*},K_{2,n},K_{2}+nK_{1},K_{1}+nK_{2}$

is $n,n-2,n+1,n+2,n+2,2n+1$ respectively. Let $n=c+3$, then all these graphs
are connected but the number of vertices with degree at least $2$ is not
bounded by $c$, so they are not $\mathcal{H}$-free. Hence
$\mathcal{H}\leq\\{K_{n},P_{n},K_{1,n}^{*},K_{2,n},K_{2}+nK_{1},K_{1}+nK_{2}\\}$.

Next we prove the “if” part. Now
$\mathcal{H}\leq\\{K_{n},P_{n},K_{1,n}^{*},K_{2,n},K_{2}+nK_{1},K_{1}+nK_{2}\\}$.
Since $\mathcal{H}$-free graph is also
$\\{K_{n},P_{n},K_{1,n}^{*},K_{2,n},K_{2}+nK_{1},K_{1}+nK_{2}\\}$-free, we
need to show that if a connected graph $G$ satisfies

$\\#\\{v\in V(G):\deg(v)\geq 2\\}>N_{0}(N_{1}),\,N_{1}=(n-1)(R_{2}(2n-1)),$

then it contains $K_{n},P_{n},K_{1,n}^{*},K_{2,n},K_{2}+nK_{1}$ or
$K_{1}+nK_{2}$ as an induced subgraph. ($N_{0}$ comes from Proposition 1.2.)
For a cut vertex $v$ of $H\leq G$, an induced subgraph of $G$, it is connected
to at least $2$ components of $H-v$, so $v$ has at least $2$ neighbors in $H$,
$\deg_{G}(v)\geq 2$. Deleting all the vertices of degree $1$ in $G$ one by
one, the left induced subgraph $G^{\prime}$ with order at least $N_{0}(N_{1})$
is still connected. So by Proposition 1.2, it contains $K_{N_{1}},P_{N_{1}}$
or $K_{1,N_{1}}$. As $N_{1}\geq n$, we are done if $G^{\prime}$ contains
$K_{N_{1}}$, or $P_{N_{1}}$.

We may assume $G^{\prime}$ contains $K=K_{1,N_{1}}$. Denote the central vertex
by $s$, the leaves by $X=\\{x_{i}:i\in[N_{1}]\\}$. Since $\deg_{G}(x_{i})\geq
2$ in $G$, we can find another vertex $y_{i}\not=s$ in $G$ adjacent to
$x_{i}$. (Note that $y_{i}$ and $y_{j}$, $i,j\in[N_{1}],\,i\not=j$ maybe the
same vertex.) If there is a $y_{i}$, satisfying $|N(y_{i})\cap X|\geq n$, then
$\\{s,y_{i},N(y_{i})\cap X\\}$ contains $K_{2,n}$ or $K_{2}+nK_{1}$ as an
induced subgraph according to whether $y_{i}$ is adjacent to $s$.

Hence we may assume $|N(y_{i})\cap X|<n,\;\forall i\in[N_{1}].$ Denote by $Y$
the set consisting of vertices $y_{i},i\in[N_{1}]$. Consider the bipartite
graph $G^{\prime}[X,Y]$. By Lemma 2.1, we can find $X^{\prime}\subset
X,\,Y^{\prime}\subset Y$ which form an induced matching of order $R_{2}(2n-1)$
in $G^{\prime}[X,Y]$. If $Y^{\prime}$ contains a clique of order $n$, then we
are done. Since $|Y^{\prime}|=R_{2}(2n-1)$, $Y^{\prime}$ contains a stable set
$S$ of order $2n-1$. Using pigeonhole principle, if $|N(s)\cap S|\geq n$, then
$G$ contains an $K_{1}+nK_{2}$ as an induced subgraph. Otherwise, $|N(s)\cap
S|<n$, there are at least $n$ vertices not adjacent to $s$, so $G$ contains
$K_{1,n}^{*}$.

In other words, every connected
$\\{K_{n},P_{n},K_{1,n}^{*},K_{2,n},K_{2}+nK_{1},K_{1}+nK_{2}\\}$-free graph
$G$ satisfies $\\#\\{v\in V(G):\deg(v)\geq 2\\}\leq c=(n-1)R_{2}(2n-1)$. The
proof is completed. ∎

###### Proof of Theorem 1.4.

For a graph $G$, the vertex $v$ such that $\alpha(N(v))=1$ is not a cut vertex
of any induced subgraph of $G$ which contains $v$. Deleting all these vertices
one by one, the left induced subgraph $G^{\prime}$ is still connected.

We first prove the “only if” part. For
$G=K_{n}^{*},P_{n},K_{1,n}^{*},K_{2,n},E_{2}+K_{n},K_{1}+nP_{3}$, or $CK_{n}$,
the order of $G^{\prime}$ is $n,n-2,n+1,n+2,n,n+1,2n$ respectively. Let
$n=c+3$, we know that
$\mathcal{H}\leq\\{K_{n}^{*},P_{n},K_{1,n}^{*},K_{2,n},E_{2}+K_{n},K_{1}+nP_{3},CK_{n}\\}$.

Next we prove the “if” part. Set

$N_{1}=2N_{2}-1,\;N_{2}=nR_{2}(n)N_{3},\;N_{3}=R_{2^{8}}(n+2).$

All we need to show is that if a connected graph $G$ satisfies

$\\#\\{v\in V(G):\alpha(N(v))\geq 2\\}>N_{0}(N_{1}),$

then it contains
$K_{n}^{*},P_{n},K_{1,n}^{*},K_{2,n},E_{2}+K_{n},K_{1}+nP_{3}$ or $CK_{n}$ as
an induced subgraph. By Proposition 1.2, $G^{\prime}$ contains
$P_{N_{1}},\;K_{N_{1}}$ or $K_{1,N_{1}}$ as an induced subgraph. If
$G^{\prime}$ contains $P_{N_{1}}$, we are done. We divide it into two cases.

Case 1: $G^{\prime}$ contains $X=K_{N_{1}}$.

All vertices in $X$ have local independence number at least $2$ in G. For a
vertex $x\in X$, if there doesn’t exist a $y\in G$ adjacent to $x$ but not
complete to $X$, then in order to ensure $\alpha(N(x))\geq 2$, there are $y$
and $z$ complete to $K$ satisfying $y$ and $z$ are not adjacent. Hence $G$
contains $E_{2}+K_{n}$.

So we may assume that for any $x\in X$, there exists a $y$ adjacent to $x$ but
not complete to $X$. Denote by $Y$ the set consisting of these corresponding
$y$. If there is a $y\in Y$ such that $|N(y)\cap X|\geq n$, then $G$ contains
$E_{2}+K_{n}$. Done. So we may assume $|N(y)\cap X|<n,\forall y\in Y$.
Applying Lemma 2.1 to bipartite graph $G[X,Y]$, we can find
$\\{x_{i},y_{i}:i\in[R_{2}(n)]\\}$ which forms an induced matching of order
$R_{2}(n)$ in $G[X,Y]$. If $Y^{\prime}=\\{y_{i}:i\in[R_{2}(n)]\\}$ contains a
clique of order $n$, then $G$ contains $CK_{n}$. Otherwise $Y^{\prime}$
contains an stable set of order $n$, then $G$ contains $K_{n}^{*}$.

Case 2: $G^{\prime}$ contains $K=K_{1,N_{1}}$.

Denote the central vertex by $s$, the set of leaves by $X$. For $x\in X$, in
order to ensure $\alpha(N(x))\geq 2$, one way is that there exists a $y$
adjacent to $x$ but not $s$, and we denote by $x\in X_{1}$. Otherwise another
way is that there exists $y\sim z$ adjacent to $x$ and $s$, and we denote by
$x\in X_{2}$. So $X$ is the disjoint union of $X_{1}$ and $X_{2}$.
$N_{1}=2N_{2}-1$, use pigeonhole principle.

Subcase 2.1: $|X_{1}|\geq N_{2}$.

Denote by $Y$ the set consisting of vertices which is the corresponding $y$
for $x\in X_{1}$. If some $y\in Y$ satisfies $|N(y)\cap X_{1}|\geq n$, then
$G$ contains $K_{2,n}$. For bipartite graph $G^{\prime}[X_{1},Y]$, we can
assume $\Delta(Y)\leq n-1$. Apply Lemma 2.1 to it, we can find
$\\{x_{i},y_{i}:i\in[R_{2}(n)]\\}$ which forms an induced matching of
$G^{\prime}[X_{1},Y]$. If $Y^{\prime}=\\{y_{i}:i\in[R_{2}(n)]\\}$ contains a
clique of order $n$, then $G$ contains $K_{n}^{*}$; otherwise $Y^{\prime}$
contains a stable set of order $n$, then $G$ contains $K_{1,n}^{*}$.

Subcase 2.2: $|X_{2}|\geq N_{2}$.

Hence we can find vertices $x_{i},y_{i},z_{i}:i\in[N_{2}]$ such that
$\\{s,x_{i},y_{i},z_{i}\\}$ forms an induced $K_{4}-e$ since $y_{i}$ is not
adjacent to $z_{i}$. Note that $y_{i}$ and $z_{i}$ are distinct and symmetric.
However, $y_{i}$ maybe the same vertex as $y_{j}$ or $z_{j}$. If some vertex
$s^{\prime}$ truly appears $nR_{2}(n)$ times, then replace $s$ by
$s^{\prime}$. We return to subcase 2.1. For example, if
$s^{\prime}=z_{i}:i\in[nR_{2}(n)]$, then
$s^{\prime},x_{i},y_{i}:i\in[nR_{2}(n)]$ is what we need in subcase 2.1.

So without loss of generality, by contracting $N_{2}$ to $N_{3}$, we can
assume $y_{i},z_{i}:i\in[N_{3}]$ are distinct vertices. Color the auxiliary
complete graph $K_{N_{3}}$. The color of edge $(i,j),\,i<j$ is

$(\mathbbm{1}_{x_{i}\sim y_{j}},\,\mathbbm{1}_{x_{i}\sim
z_{j}},\,\mathbbm{1}_{y_{i}\sim x_{j}},\,\mathbbm{1}_{z_{i}\sim
x_{j}},\,\mathbbm{1}_{y_{i}\sim y_{j}},\,\mathbbm{1}_{z_{i}\sim
z_{j}},\,\mathbbm{1}_{y_{i}\sim z_{j}},\,\mathbbm{1}_{z_{i}\sim y_{j}}).$

$N_{3}=R_{2^{8}}(n+2)$, so there is a monochromatic clique $K$ of order $n+2$.
Without loss of generality, we assume $V(K)=[n+2]$. Now we consider the color
of $K$.

For $i,j\in[n+2],\,i<j$, suppose $\mathbbm{1}_{y_{i}\sim y_{j}}=1$ first. If
$\mathbbm{1}_{x_{i}\sim y_{j}}=1$ then $\\{x_{1},x_{2},y_{i}:i\in[3,n+2]\\}$
forms an induced $E_{2}+K_{n}$. By symmetry, ($y_{i}$ and $z_{i}$ are
symmetric, $i$ and $j$ are symmetric, e.g. the arguments for
$\mathbbm{1}_{x_{i}\sim y_{j}}$ and $\mathbbm{1}_{y_{i}\sim x_{j}}$ are
symmetric.) otherwise $\mathbbm{1}_{x_{i}\sim y_{j}}=\mathbbm{1}_{y_{i}\sim
x_{j}}=0$, $\\{x_{i},y_{i}:i\in[n]\\}$ forms an induced $K_{n}^{*}$.

Next we can assume $\mathbbm{1}_{y_{i}\sim y_{j}}=\mathbbm{1}_{z_{i}\sim
z_{j}}=0$. If $\mathbbm{1}_{x_{i}\sim y_{j}}=1$, then
$\\{x_{1},x_{2},y_{i}:i\in[3,n+2]\\}$ forms an induced $K_{2,n}$. If
$\mathbbm{1}_{y_{i}\sim z_{j}}=1$, then $\\{y_{1},y_{2},z_{i}:i\in[3,n+2]\\}$
forms an induced $K_{2,n}$. By symmetry, now we can assume

$\mathbbm{1}_{x_{i}\sim y_{j}}=\mathbbm{1}_{y_{i}\sim
x_{j}}=\mathbbm{1}_{x_{i}\sim z_{j}}=\mathbbm{1}_{z_{i}\sim
x_{j}}=\mathbbm{1}_{y_{i}\sim z_{j}}=\mathbbm{1}_{z_{i}\sim y_{j}}=0,$

hence $\\{s,x_{i},y_{i},z_{i}:i\in[n]\\}$ forms an induced $K_{1}+nP_{3}$. ∎

Furuya [8] considered the Ramsey-type problem for the parameter domination
number. Actually it got the following result.

######  Theorem 2.2.

[8] The following are equivalent:

(1) There is a constant $c=c(\mathcal{H})$ such that the connected domination
number $\gamma_{c}(G)<c$ for every connected $\mathcal{H}$-free graph $G$.

(2) There is a constant $c=c(\mathcal{H})$ such that the domination number
$\gamma(G)<c$ for every connected $\mathcal{H}$-free graph $G$.

(3) $\mathcal{H}\leq\\{K_{1,n}^{*},K_{n}^{*},P_{n}\\}$ for some positive
integer $n$.

###### Proof of Theorem 1.5.

We first prove the “only if” part. For graph $G$, set $G^{\prime}=\\{v\in
V(G):c(N(v))\geq 2\\}$. For $G=K_{n}^{*},P_{n},K_{1,n}^{*},K_{2,n},CK_{n}$ or
$T_{n}$, the corresponding $G^{\prime}$ has order $n,n-2,n+1,n+2,2n,n+1$
respectively. Let $n=c+3$, then

$\mathcal{H}\leq\\{K_{n}^{*},P_{n},K_{1,n}^{*},K_{2,n},CK_{n},T_{n}\\}.$

Next we prove the “if” part. By Theorem 2.2, connected
$\\{K_{n}^{*},P_{n},K_{1,n}^{*}\\}$-free graph $G$ has a uniform bound
$\gamma_{n}$ such that $\gamma(G)\leq\gamma_{n}$. All we need to show is that
if a connected graph $G$ satisfies

$\\#\\{v\in V(G):c(N(v))\geq
2\\}>\gamma_{n}R_{2}(N_{1}),\,N_{1}=nN_{2},\,N_{2}=R_{2}(n),$

then it contains $K_{n}^{*},P_{n},K_{1,n}^{*},K_{2,n},T_{n}$ or $CK_{n}$ as an
induced subgraph.

Take a minimum dominating set $D$ of $G$, $|D|\leq\gamma_{n}$,
$|G^{\prime}|>\gamma_{n}R_{2}(N_{1})$, so

$|G^{\prime}-D|>\gamma_{n}(R_{2}(N_{1})-1).$

Hence there is a $d\in D$, such that $|N(d)\cap G^{\prime}|\geq R_{2}(N_{1})$.
If $N(d)\cap G^{\prime}$ contains a clique of order $N_{1}$, then we have a
$K_{N_{1}}$ in $G^{\prime}$. Otherwise, $N(d)\cap G^{\prime}$ contains a
stable set of order $N_{1}$, then we have a $K_{1,N_{1}}$ which leaves are in
$G^{\prime}$.

Every vertex $v$ in $G^{\prime}$ satisfies $c(N(v))\geq 2$ in $G$, so we can
find more vertices.

Case 1: $G^{\prime}$ contains $K=K_{N_{1}}=\\{x_{i}:i\in[N_{1}]\\}$. So there
is a $y_{i}$ adjacent to $x_{i}$ such that $y_{i}$ can not be connected to
$K-x_{i}$ in $N(x_{i})$. Hence $y_{i}\not=y_{j}$ if
$i\not=j,\,\forall\,i,j\in[N_{1}]$. $N_{1}\geq R_{2}(n)$. If
$Y=\\{y_{i}:i\in[N_{1}]\\}$ contains a clique of order $n$, then $G$ contains
$CK_{n}$; otherwise $Y$ contains a stable set of order $n$, then $G$ contains
$K_{n}^{*}$.

Case 2: There is a $K=K_{1,N_{1}}=\\{s,\,x_{i}:i\in[N_{1}]\\}$ such that $s$
is the central vertex, and $x_{i}\in G^{\prime},\forall i\in[N_{1}]$. Hence
there is a $y_{i}$ adjacent to $x_{i}$ but not $s$. If for some $i$,
$|N(y_{i})\cap K|\geq n$, then $G$ contains $K_{2,n}$, done. Otherwise, by
Lemma 2.1, contracting $N_{1}$ into $N_{2}$, we can assume $N(y_{i})\cap
K=\\{x_{i}\\}$. $N_{2}=R_{2}(n)$. If $Y=\\{y_{i}:i\in[N_{2}]\\}$ contains
$K_{n}$, then $G$ contains $T_{n}$. Otherwise $Y$ contains $E_{n}$, hence $G$
contains $K_{1,n}^{*}$. ∎

###### Proof of Theorem 1.6.

We first prove the “only if” part. The number of cut vertices for
$K_{n}^{*},P_{n},K_{1,n}^{*}$ are $n,n-2$ and $n+1$ respectively. Take
$n=c+3$, then $K_{n}^{*},P_{n},K_{1,n}^{*}$ are connected and don’t satisfy
the conclusion, so they are not $\mathcal{H}$-free, hence
$\mathcal{H}\leq\\{K_{n}^{*},K_{1,n}^{*},P_{n}\\}$.

Next we prove the “if” part. By Theorem 2.2, there is a constant
$\gamma_{c}(n)$ such that every connected
$\\{K_{n}^{*},K_{1,n}^{*},P_{n}\\}$-free graph $G$ satisfies its connected
domination number $\gamma_{c}(G)\leq\gamma_{c}(n)$.

######  Lemma 2.3.

For a connected graph $G$, the connected dominating set $D$ contains all cut
vertices of $G$.

###### Proof.

Suppose not, then there is a cut vertex $v$ of $G$ such that $v\not\in D$.
$G-v$ is not connected. $D\subset G-v$ is connected. So there is a connected
component $C$ of $G-v$ which stays away from $D$. This contradicts to that $D$
is a dominating set. ∎

So if $\mathcal{H}\leq\\{K_{n}^{*},K_{1,n}^{*},P_{n}\\}$, then every connected
$\mathcal{H}$-free graph $G$ is also $\\{K_{n}^{*},K_{1,n}^{*},P_{n}\\}$-free,
the number of cut vertices of $G$ is bounded by $\gamma_{c}(n)$.

∎

##  3 Graph with many vertices of large parameter

The following lemma is similar to the Ramsey’s theorem. As for proof, we may
assume $k=2$, and consider the bipartite case. Then use the same technique as
the classical proof of Ramsey’s theorem.

######  Lemma 3.1.

[2] For any positive integer $k$ and $q$, there exists a number $MR(k,q)$ such
that in every $k$-partite graph $G=(V_{1},V_{2},\dots,V_{k},E)$ with
$|V_{i}|\geq MR(k,q),\forall i$, there is a collection of subsets
$U_{i}\subset V_{i}$ of order $|U_{i}|=q$ satisfying every pair of subsets
induces either a biclique $K_{q,q}$ or its bipartite complement $E_{2q}$.

###### Proof of Theorem 1.8.

We first prove the “only if” part. Set $n=c_{1}+c_{2}$, then for
$G=K_{n},K_{n,n},nK_{1,n}$, the number of vertices with degree at least
$c_{1}$ is $n,2n,n$ respectively. So they are not $\mathcal{H}$-free, hence
$\mathcal{H}\leq\\{K_{n},K_{n,n},nK_{1,n}\\}$.

Next we prove the “if” part.

Claim: Set $N_{3}=R_{2}(n),\;N_{2}=R_{2^{n^{2}+2n+1}}(2n),\;N_{1}=N_{2}\cdot
N_{3}+N_{2}$, then for every $\\{K_{n},K_{n,n},nK_{1,n}\\}$-free graph $G$, we
have

$\\#\\{v\in V(G):\deg(v)\geq N_{1}\\}<N_{2}.$

Suppose not, name the vertices with large degree by $v_{i},i\in[N_{2}]$.
$\deg(v_{i})\geq N_{2}\cdot N_{3}+N_{2}$, so for $v_{i}$, we can find
$v_{i}^{j}:j\in[N_{3}]$ adjacent to $v_{i}$. All
$v_{i}^{j}:i\in[N_{2}],\,j\in[N_{3}]$ are distinct even though $v_{i}$ is
adjacent to $v_{i^{\prime}}$ and $v_{i^{\prime}}^{j}$. For every $i$, we can
find $n$ stable vertices from $\\{v_{i}^{j}:j\in[N_{3}]\\}$ since there is no
$K_{n}$. For convenience, we denote these $n$ stable vertices by
$\\{v_{i}^{j}:j\in[n]\\}$. Color the auxiliary complete graph $K_{N_{2}}$. The
edge $(i,i^{\prime}),\,i<i^{\prime}$ is colored by

$(\mathbbm{1}_{v_{i}\sim v_{i^{\prime}}},\mathbbm{1}_{v_{i}\sim
v_{i^{\prime}}^{j}},\mathbbm{1}_{v_{i^{\prime}}\sim
v_{i}^{j}},\mathbbm{1}_{v_{i}^{j}\sim
v_{i^{\prime}}^{j^{\prime}}}:j,j^{\prime}\in[n])$

We use $2^{n^{2}+2n+1}$ colors, $N_{2}=R_{2^{n^{2}+2n+1}}(2n)$, so we can find
a monochromatic clique $K$ of order $2n$. For convenience, suppose
$V(K)=[2n]$. Consider the color of $K$.

$G$ is $K_{n}$-free, so $\mathbbm{1}_{v_{i}\sim
v_{i^{\prime}}}=\mathbbm{1}_{v_{i}^{j}\sim v_{i^{\prime}}^{j}}=0$. If
$\mathbbm{1}_{v_{i}^{j}\sim v_{i^{\prime}}^{j^{\prime}}}=1$ for some distinct
$j,j^{\prime}\in[n]$ then
$\\{v_{1}^{j},\dots,v_{n}^{j};v_{n+1}^{j^{\prime}},\dots,v_{2n}^{j^{\prime}}\\}$
forms a $K_{n,n}$, contradiction. So $\mathbbm{1}_{v_{i}^{j}\sim
v_{i^{\prime}}^{j^{\prime}}}=0$. If $\mathbbm{1}_{v_{i}\sim
v_{i^{\prime}}^{j}}=1$ for some $j$, then
$\\{v_{1},\dots,v_{n};v_{n+1}^{j},\dots,v_{2n}^{j}\\}$ forms a $K_{n,n}$, a
contradiction. By symmetry, all the characteristic function is $0$. Thus we
have a $nK_{1,n}$.

Therefore the claim is true. Set $c_{1}=N_{2}\cdot N_{3}+N_{2},\;c_{2}=N_{2}$,
we complete the proof. ∎

###### Proof of Theorem 1.9.

It’s clear that “$(1)\Rightarrow(2)$” holds sine $\alpha(N(v))\geq c(N(v))$.

We show that “$(2)\Rightarrow(3)$” holds. Set $n=c_{1}+c_{2}$, then
$\\#\\{v\in V(G):c(N(v))\geq c_{1}\\}$ for
$G=K_{n,n},nK_{1,n},K_{n}+E_{n},K_{n}^{n}$ is $2n,n,n,n$ respectively. All are
larger than $c_{2}$. So they are not $\mathcal{H}$-free. Hence
$\mathcal{H}\leq\\{K_{n,n},nK_{1,n},K_{n}+E_{n},K^{n}_{n}\\}$.

Finally We show that “$(3)\Rightarrow(1)$” holds. Set

$N_{2}=R_{2^{2n+1}}(2n),\;N_{3}=MR(N_{2},3n),\;N_{1}=N_{2}\cdot N_{3}+N_{2}.$

We only need to show that if a graph $G$ satisfies

$\\#\\{v\in V(G):\alpha(N(v))\geq c_{1}=N_{1}\\}\geq c_{2}=N_{2}$

then it contains $K_{n,n},nK_{1,n},K_{n}+E_{n}$ or $K^{n}_{n}$.

Take $N_{2}$ vertices $v_{i},i\in[N_{2}]$ satisfying $\alpha(N(v_{i}))\geq
c_{1}$. So we can find $N_{3}$ stable neighbors of $v_{i}$ separately. By
Lemma 3.1, since $N_{3}=MR(N_{2},3n)$, we can require all these neighbors of
$v_{i},i\in[N_{2}]$ are stable, i.e. we have stable set
$\\{v_{i}^{j}:i\in[N_{2}],j\in[3n]\\}$ satisfying $v_{i}\sim v_{i}^{j}$. Color
the auxiliary complete graph $K_{N_{2}}$. The edge
$(i,i^{\prime}),\,i<i^{\prime}$ is colored by

$(\mathbbm{1}_{v_{i}\sim v_{i^{\prime}}},\mathbbm{1}_{v_{i}\sim
v_{i^{\prime}}^{j}},\mathbbm{1}_{v_{i^{\prime}}\sim v_{i}^{j}}:j\in[3n]).$

Then we can get a monochromatic clique $K$ of order $2n$. Without loss of
generality, we may assume $V(K)=[2n]$. We now handle the color of $K$. Suppose
firstly $\mathbbm{1}_{v_{i}\sim v_{i^{\prime}}}=0$. If $\mathbbm{1}_{v_{i}\sim
v_{i^{\prime}}^{j}},\mathbbm{1}_{v_{i^{\prime}}\sim v_{i}^{j}}:j\in[3n]$ are
all $0$, then $G$ contains $nK_{1,n}$. Otherwise, by symmetry, there exists
some $j$ such that $\mathbbm{1}_{v_{i}\sim v_{i^{\prime}}^{j}}=1$, then
$\\{v_{i}:i\in[n],v_{i^{\prime}}^{j}:i^{\prime}\in[n+1,2n]\\}$ forms a
$K_{n,n}$.

Suppose now $\mathbbm{1}_{v_{i}\sim v_{i^{\prime}}}=1$. If
$J=\\{j\in[3n]:\mathbbm{1}_{v_{i}\sim v_{i^{\prime}}^{j}}=1\\}$ has more than
$n-1$ elements, ($|J|\geq n$.) then $\\{v_{i}:i\in[n],v_{n+1}^{j}:j\in J\\}$
contains a $K_{n}+E_{n}$. By symmetry, we assume $J$ and
$J^{\prime}=\\{j\in[3n]:1_{v_{i^{\prime}}\sim v_{i}^{j}}=1\\}$ have no more
than $n-1$ elements. So every $v_{i}$ have $n$ leaves not adjacent to other
$v_{k},\,\forall k\not=i$. Hence $G$ contains a $K_{n}^{n}$ as an induced
subgraph. ∎

###### Proof of Theorem 1.10.

We first prove the “only if” part. Set $n=c_{1}+c_{2}$, then for
$G=nK_{1,n},K^{n}_{n}$, the number of vertices with adhesion at least $c_{1}$
is $n,n$ respectively. So they are not $\mathcal{H}$-free, hence
$\mathcal{H}\leq\\{nK_{1,n},K^{n}_{n}\\}$.

Next we prove the “if” part.

Claim: Set $N_{2}=R_{2}(n),\;N_{3}=n+N_{2}-1,\;N_{1}=N_{2}\cdot N_{3}+N_{2}$,
then for every $\\{nK_{1,n},K_{n}^{n}\\}$-free graph $G$, we have

$\\#\\{v\in V(G):\operatorname{adh}(v)\geq N_{1}\\}<N_{2}.$

Suppose not, we can find $N_{2}$ vertices $\\{v_{i}:i\in[N_{2}]\\}$ of large
adhesion. Hence there is $\\{v_{i}^{j}:j\in[N_{3}]\\}\subset N(v_{i})$ for any
$i$ such that $v_{i}^{j}$ can be connected to $v_{i}^{j^{\prime}}$ only
through $v_{i}$. If there are two vertices $v_{i}^{j}$ and
$v_{i}^{j^{\prime}}$ satisfying
$N(v_{i}^{j})\cap\\{v_{i^{\prime}},v_{i^{\prime}}^{k}:k\in[N_{3}]\\}\not=\emptyset$
and
$N(v_{i}^{j^{\prime}})\cap\\{v_{i^{\prime}},v_{i^{\prime}}^{k}:k\in[N_{3}]\\}\not=\emptyset$,
then it is contradicted to that $v_{i}^{j}$ can be connected to
$v_{i}^{j^{\prime}}$ only through $v_{i}$. So by change $N_{3}$ into
$N_{3}-(N_{2}-1)=n$, we may assume

$N(v_{i}^{j})\cap\\{v_{i^{\prime}},v_{i^{\prime}}^{k}:k\in[n]\\}=\emptyset,\forall
i\not=i^{\prime},\forall j\in[n].$

The set $V=\\{v_{i}:i\in[N_{2}]\\}$ has order $N_{2}=R_{2}(n)$. So if $V$ has
a clique of order $n$, then $G$ contains $K_{n}^{n}$, otherwise $V$ has a
stable set of order $n$, then $G$ contains $nK_{1,n}$. The claim is complete.
Set $c_{1}=N_{1},\,c_{2}=N_{2}$, the proof is complete. ∎

##  4 Concluding remarks

In Theorem 1.6, we require that the $\mathcal{H}$-free graph $G$ is connected.
If we drop this condition, we can find stronger $\mathcal{H}$, so the problem
will be easier. We have the following corollary.

######  Corollary 4.1.

There is a constant $c=c(\mathcal{H})$ such that $\\#\\{v\in V(G):\deg(v)\geq
2\\}\leq c$ for every $\mathcal{H}$-free graph $G$ if and only if
$\mathcal{H}\leq\\{K_{n},nP_{3},nK_{3},K_{1,n}^{*},K_{2,n},K_{2}+nK_{1},K_{1}+nK_{2}\\}$
for some $n$.

###### Proof.

Compare with the corresponding connected version. We first prove the “only if”
part.

Suppose $\mathcal{H}$ satisfies there is a constant $c=c(\mathcal{H})$ such
that $\\#\\{v\in V(G):\deg(v)\geq 2\\}\leq c$ for every $\mathcal{H}$-free
graph $G$. The number of vertices with degree at least $2$ for

$K_{n},nP_{3},nK_{3},K_{1,n}^{*},K_{2,n},K_{2}+nK_{1},K_{1}+nK_{2}$

are $n,n,3n,n+1,n+2,n+2,2n+1$ respectively. Let $n=c+1$, then all these graphs
satisfy the number of vertices with degree at least $2$ is not bounded by $c$,
so they are not $\mathcal{H}$-free. Hence
$\mathcal{H}\leq\\{K_{n},nP_{3},nK_{3},K_{1,n}^{*},K_{2,n},K_{2}+nK_{1},K_{1}+nK_{2}\\}$.

Next we prove the “if” part. Suppose

$\mathcal{H}\leq\\{K_{n},nP_{3},nK_{3},K_{1,n}^{*},K_{2,n},K_{2}+nK_{1},K_{1}+nK_{2}\\}.$

Then every $\mathcal{H}$-free graph $G$ is also
$\\{K_{n},nP_{3},nK_{3},K_{1,n}^{*},K_{2,n},K_{2}+nK_{1},K_{1}+nK_{2}\\}$-free.
The connected component of $G$ which contains a vertex of degree larger than
$2$ must contain an induced $P_{3}$ or $K_{3}$. Since $G$ is
$\\{nP_{3},nK_{3}\\}$-free, there are at most $2n-2$ connected components of
$G$ which contains a vertex of degree larger than $1$. For a connected
component $C$, since $C$ is
$\\{K_{4n},P_{4n},K_{1,4n}^{*},K_{2,4n},K_{2}+4nK_{1},K_{1}+4nK_{2}\\}$-free,
by Theorem 1.3,

$\\#\\{v\in V(C):\deg(v)\geq 2\\}\leq
N_{0}(N_{1}),\,N_{1}=(4n-1)(R_{2}(8n-1)),$

hence $\\#\\{v\in V(G):\deg(v)\geq 2\\}\leq(2n-2)N_{0}(N_{1})$. Done. ∎

Use the similar technique, we can prove the following.

######  Corollary 4.2.

There is a constant $c=c(\mathcal{H})$ such that

$\\#\\{v\in V(G):\alpha(N(v))\geq 2\\}\leq c$

for every $\mathcal{H}$-free graph $G$ if and only if

$\mathcal{H}\leq\\{K_{n}^{*},nP_{3},K_{1,n}^{*},K_{2,n},E_{2}+K_{n},CK_{n}\\}$

for some $n$.

######  Corollary 4.3.

There is a constant $c=c(\mathcal{H})$ such that

$\\#\\{v\in V(G):c(N(v))\geq 2\\}\leq c$

for every $\mathcal{H}$-free graph $G$ if and only if

$\mathcal{H}\leq\\{K_{n}^{*},nP_{3},K_{1,n}^{*},K_{2,n},CK_{n},T_{n}\\}$

for some $n$.

######  Corollary 4.4.

There is a constant $c=c(\mathcal{H})$ such that

$\\#\\{v\in V(G):\operatorname{adh}(v)\geq 2\\}\leq c$

for every $\mathcal{H}$-free graph $G$ if and only if
$\mathcal{H}\leq\\{K_{n}^{*},nP_{3},K_{1,n}^{*}\\}$ for some $n$.

Note that the proof previously appeared can be adjusted slightly and the bound
can be made smaller. For example, we can use Theorem 2.2 to handle theorems in
section $2$ as the same way as Theorem 1.5. We can require that the graphs in
$\mathcal{H}$ use different positive integer, such as
$\mathcal{H}\leq\\{K_{1,n_{1}},K_{n_{2}},P_{n_{3}}\\}$ in Theorem 1.6. We can
also require that the constant $c_{1}(\mathcal{H})$ and $c_{2}(\mathcal{H})$
be the same in Theorem 1.8 since we can replace $c_{1}(\mathcal{H})$ and
$c_{2}(\mathcal{H})$ by the same constant
$c_{1}(\mathcal{H})+c_{2}(\mathcal{H})$.

However, it has no meaning except making proofs seem more complicated. As for
Theorem 1.8, Theorem 1.9 and Theorem 1.10, we don’t require $G$ is connected,
because there is no typical way to connect the $n$ discrete $K_{1,n}$. For a
tree with many leaves, replace the leaves by $K_{1,n}$, then we get a
counterexample. However, we have no good way to forbid this case. The more
explanation, see [4], [10].

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