details

# Texture-induced spin-orbit coupling and Skyrmion-electron bound states
in a Néel antiferromagnet

N. Davier<EMAIL_ADDRESS>Laboratoire de Physique Théorique,
Université de Toulouse, CNRS, UPS, France  R. Ramazashvili<EMAIL_ADDRESS>tlse.fr Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS,
France

###### Abstract

We derive effective-mass electron Hamiltonian for a Néel antiferromagnet in
the presence of a smooth texture of the staggered magnetization. For certain
locations of electron band extrema, the texture produces a peculiar and
anomalously strong spin-orbit coupling of the scale $\hbar v/L$, with $v$ the
Fermi velocity and $L$ the characteristic length scale of the texture. For a
Skyrmion texture, this coupling generates electron bound states, whose energy
scale is given by the gap $\Delta$ in the electron spectrum. With dopant
carriers, such bound states turn the Skyrmion into a charged particle, that
can be manipulated by electric field.

##  I Introduction

It is an honor and our great pleasure to contribute to this Festschrift for
the 95-th birthday of Emmanuel Rashba. One of us (R.R.) first came under the
spell of E. I. Rashba’s papers as a student, and turns to them for
understanding and inspiration to this day. This modest offering is our ‘thank
you’. Happy birthday, Emmanuel Iosifovich, many happy returns of the day!

Topological textures such as domain walls, vortices and Skyrmions appear
prominently in diverse areas of physics, from cosmology and string theory
Schwarz  to QCD, the physics of hadrons, and condensed matter physics Rho . In
solid state magnetism alone, topological textures bring together fundamental
and applied science, from novel states of matter such as Skyrmion crystal to
prototype spintronic devices that employ Skyrmions and domain walls to process
information Back-2020 . While early spintronics research largely focussed on
ferromagnetic materials Fert-2008 ; Gruenberg-2008 ; Fert-2019 , an ever
increasing effort has been turning to antiferromagnets Gomonay-2017 ;
Baltz-2018  and to topological textures therein Smejkal-2018 ; Goebel-2021  –
in view of their technologically attractive properties such as shorter
characteristic timescales of the Néel state and its lack of net magnetization
Jungwirth-2016 .

In order to usefully employ topological textures, it is crucial to understand
their interplay with other subsystems of a solid, first and foremost – with
band electrons. For ferromagnets, much work has been done to understand the
influence of a texture on electric current and vice versa Tatara-2008 ;
Nagaosa-2013 . Soon after, this effort has been expanded to antiferromagnets
MacDonald-2011 ; Gomonay-2014 ; Baltz-2018 ; Manchon-2019 .

An important step in this direction has been undertaken by R. Cheng and Q. Niu
Cheng-2012 , who studied electron motion in the presence of a texture in a
Néel antiferromagnet, using coupled quasiclassical equations of motion for the
electron momentum, coordinate and spin. Here we address the same problem fully
quantum-mechanically, by deriving the effective-mass Hamiltonian – and show
how, for certain locations of electron band extrema, the texture generates a
peculiar and anomalously strong spin-orbit coupling, a key result of this
work.

It is at this point that we find ourselves following in the footsteps of
Solomon Pekar and Emmanuel Rashba Pekar-1964 . Back in 1964, Pekar and Rashba
observed that a non-uniform magnetic field as well as non-uniform
magnetization $\mathcal{M}({\bf r})$ give rise to a spin-orbit coupling:
exchange interaction $\mathcal{M}({\bf r})\cdot{\bm{\sigma}}$ couples the
electron coordinate ${\bf r}$ to its spin ${\bm{\sigma}}$ via ${\bf
r}$-dependence of $\mathcal{\bm{M}}({\bf r})$ – that is, simply by virtue of
inhomogeneity.

The observation above appears to suggest non-degenerate bands, split by
exchange $\mathcal{M}({\bf r})\cdot{\bm{\sigma}}$. Yet symmetry can void this
argument by restoring the double degeneracy, as it does for a centrosymmetric
Néel antiferromagnet in its uniform state: here, magnetization
$\mathcal{\bm{M}}({\bf r})$ changes sign upon translation ${\bf T}_{a}$ by a
lattice period $a$ as well as upon time reversal $\theta$. As a result,
combined anti-unitary symmetry $I\theta{\bf T}_{a}$, with $I$ the inversion
operator, guarantees double degeneracy of Bloch eigenstates throughout the
Brillouin zone Herring .

A localized texture breaks such a degeneracy-protecting symmetry. Below we
illustrate this by a texture in a Néel antiferromagnet, and by an unusual
spin-orbit coupling that it produces. Moreover, for a Skyrmion texture, this
spin-orbit coupling creates electron bound states with energy scale given by
the gap $\Delta$ in the electron spectrum. With dopant carriers, such bound
states turn the Skyrmion into a charged particle, another key result of this
work.

The paper is organized as follows. In Section II, we begin by deriving the
low-energy electron Hamiltonian for a centrosymmetric Néel antiferromagnet in
the presence of a texture. Then we specify the Hamiltonian for a particular
location of the electron band extrema – and point out the appearance of a
peculiar spin-orbit coupling. In Section III, we present our main example: a
texture in the form of a single Belavin-Polyakov Skyrmion Belavin-1975 ;
Rajaraman . We show that the texture-induced spin-orbit coupling produces
Skyrmion-electron bound states, and study their evolution as a function of the
Skyrmion radius. In Section IV, we discuss the implications of our results in
the light of the pioneering work Pekar-1964  of Pekar and Rashba. Finally, the
Appendices discuss the connection of our results with some of the earlier
studies, and outline the validity range of the approximations we used.

##  II Electron in the presence
of a texture

Consider a Néel antiferromagnet on a square-symmetry lattice with period $a$.
In the uniform state, its ordered moment changes sign upon elementary
translation, and couples electron states at any two momenta ${\bf p}$ and
${{\bf p}+{\bf Q}}$, separated by the Néel wave vector ${\bf
Q}=(\pm\frac{\pi}{a},\pm\frac{\pi}{a})$. The coupling has the form of exchange
$({\bm{\Delta}}\cdot{\bm{\sigma}})$, with ${\bm{\Delta}}$ proportional to the
staggered magnetization, and ${\bm{\sigma}}$ the triad of Pauli matrices,
representing electron spin. Since ${\bf p}$ and ${\bf p}+2{\bf Q}$ are
equivalent in the Brillouin zone (BZ), the Hamiltonian $\mathcal{H}$ can be
written as acting on a bispinor $\Psi=(\psi_{\bf p},\psi_{\bf p+Q})$
KulTug.1984 :

$\mathcal{H}=\left[\begin{array}[]{cc}\varepsilon({\bf
p})&({\bm{\Delta}}\cdot{\bm{\sigma}})\\\
({\bf\Delta}\cdot{\bm{\sigma}})&\varepsilon({\bf p+Q})\end{array}\right],$ (1)

where $\varepsilon({\bf p})$ is the electron dispersion in the absence of Néel
order. The spectrum $E_{\bf p}$ of $\mathcal{H}$ is doubly-degenerate, $E_{\bf
p}=\varepsilon_{+}({\bf
p})\pm\sqrt{|{\bm{\Delta}}|^{2}+\varepsilon_{-}^{2}({\bf p})}$, where
$\varepsilon_{\pm}({\bf p})\equiv\frac{1}{2}\left[\varepsilon({\bf
p})\pm\varepsilon({\bf p+Q})\right]$; it has a gap $\Delta=|{\bm{\Delta}}|$,
which turns a half-filled metal into an insulator.

In the presence of a texture ${\bm{\Delta}}_{\bf r}=\hat{\bf n}_{\bf
r}\Delta$, unit vector $\hat{\bf n}_{\bf r}$ becomes a smooth function of the
coordinate ${\bf r}$, and carriers near the extrema of $E_{\bf p}$ at momenta
${\bf p}_{0}$ and ${{\bf p}_{0}+{\bf Q}}$ admit a low-energy effective-mass
description Kittel . To derive it, in Eq. (1) we replace uniform
${\bm{\Delta}}$ by ${\bm{\Delta}}_{\bf r}$, and substitute $\hat{\bf
p}\equiv-i\hbar{\bm{\nabla}}$ for the momentum dependences $\varepsilon_{{\bf
p}_{0}}(\hat{\bf p})$ and $\varepsilon_{{\bf p}_{0}+{\bf Q}}(\hat{\bf p})$ of
$\varepsilon({\bf p})$ near ${\bf p}_{0}$ and ${\bf p}_{0}+{\bf Q}$.

Now perform a spin rotation $U_{\bf r}$ that makes ${\bm{\Delta}}_{\bf r}$
uniform: $U_{\bf r}^{\dagger}(\hat{\bf n}_{\bf r}\cdot{\bm{\sigma}})U_{\bf
r}=\sigma_{z}$ Volovik-1987 . This generates a Peierls substitution
$\hat{p}_{i}\rightarrow\hat{p}_{i}+({\bf A}_{i}\cdot{\bm{\sigma}})$ in
$\varepsilon_{{\bf p}_{0}}(\hat{\bf p})$ and $\varepsilon_{{\bf p}_{0}+{\bf
Q}}(\hat{\bf p})$, with $({\bf
A}_{i}\cdot{\bm{\sigma}})=A_{i}^{\alpha}\sigma_{\alpha}=-i\hbar U_{\bf
r}^{\dagger}\partial_{i}U_{\bf r}$. Vector potential $A_{i}^{\alpha}$ carries
real-space indices $i=x,y$ and spin indices $\alpha=x,y,z$. While different
components of $({\bf A}\cdot{\bm{\sigma}})$ do not commute, $U_{\bf r}$ is
defined only up to a non-uniform spin rotation $V^{z}_{\bf
r}=e^{i\sigma_{z}\chi}$ around $\hat{z}$: $U_{\bf r}\rightarrow U_{\bf
r}V^{z}_{\bf r}$, which is an abelian transformation. This gauge
transformation acts on $({\bf A}\cdot{\bm{\sigma}})$ in a peculiar way,
elucidated by first-order expansion in infinitesimal $\chi$:

$\delta({\bf
A}_{i}\cdot{\bm{\sigma}})=\hbar\sigma_{z}\partial_{i}\chi+\chi\left[({\bf
A}_{i}\cdot{\bm{\sigma}}),\sigma_{z}\right].$ (2)

That is, $A_{i}^{z}$ transforms as electromagnetic vector potential ($\delta
A_{i}^{z}=\hbar\partial_{i}\chi$), while ${\bf
A}_{i}^{\|}=(A_{i}^{x},A_{i}^{y})$ rotates around $\hat{z}$ by angle $2\chi$.
This observation will prove useful below.

Next, we split the bispinor $\Psi$ into two spin-$\frac{1}{2}$ components, for
states at energies near $\pm\Delta$, respectively – and thus take the $4\times
4$ (‘Dirac’) Hamiltonian (1) to its $2\times 2$ (‘Pauli-Schrödinger’) low-
energy limit LL-IV ; Ryder . Here, we focus on the conduction band (energies
$E$ near $+\Delta$) and, to first order in ${\frac{E-\Delta}{\Delta}\ll 1}$,
find the effective-mass Hamiltonian $\mathcal{H}_{{\bf p}_{0}}$ near ${\bf
p}_{0}$, with $\bar{\varepsilon}_{{\bf p}_{0}+{\bf Q}}(\hat{\bf
p})\equiv\sigma_{z}\varepsilon_{{\bf p}_{0}+{\bf Q}}(\hat{\bf p})\sigma_{z}$:

$\mathcal{H}_{{\bf p}_{0}}=\frac{\varepsilon_{{\bf p}_{0}}(\hat{\bf
p})+\bar{\varepsilon}_{{\bf p}_{0}+{\bf Q}}(\hat{\bf
p})}{2}+\frac{\left[\varepsilon_{{\bf p}_{0}}(\hat{\bf
p})-\bar{\varepsilon}_{{\bf p}_{0}+{\bf Q}}(\hat{\bf
p})\right]^{2}}{8\Delta}.$ (3)

Figure 1:  The Brillouin zone (BZ) of a square-lattice Néel antiferromagnet
with wave vector $\vec{\bf Q}=(\pm\frac{\pi}{a},\pm\frac{\pi}{a})$. The large
square shows the BZ in the paramagnetic state, the shaded square depicts the
Brillouin zone in the Néel state (MBZ). The band extrema are assumed to lie at
face centers $\Sigma_{1}$-$\Sigma_{4}$ of the MBZ. The $p_{x},p_{y}$ are the
local momentum axes near $\Sigma_{1}$, as used in the main text. The ellipses
sketch the equal-$E_{\bf p}$ lines near $\Sigma_{1}$-$\Sigma_{4}$. The bold
arrows centered at $\Sigma_{1}$-$\Sigma_{4}$ show the electron spin
polarization of low-energy conduction-band bound states in each valley, for a
large BP Skyrmion of Néel type (see main text).

The explicit form of Hamiltonian (3) depends on that of $\varepsilon_{{\bf
p}_{0}}(\hat{\bf p})$ and $\varepsilon_{{\bf p}_{0}+{\bf Q}}(\hat{\bf p})$, in
its turn defined by the symmetry of momenta ${\bf p}_{0}$ and ${\bf
p}_{0}+{\bf Q}$ in the BZ. Hereafter we focus on the extrema at midpoints
$\Sigma$ of the magnetic Brillouin zone (MBZ) boundary in Fig. 1. At point
$\Sigma$, the momentum expansion of $\varepsilon_{{\bf p}_{0}}(\hat{\bf p})$
and $\varepsilon_{{\bf p}_{0}+{\bf Q}}(\hat{\bf p})$ begins with $\pm{\bf
v}\cdot\hat{\bf p}+\hat{p}_{i}^{2}/2m_{i}$, with the paramagnetic-state Fermi
velocity ${\bf v}$ at $\Sigma$ pointing along the local $p_{y}$ in Fig. 1, and
$m_{i}=m_{x},m_{y}$ the paramagnetic-state effective masses along and normal
to the MBZ boundary. Truncating the momentum expansion of Eq. (3) at quadratic
terms footnote:re-Hamiltonian , we find

$\mathcal{H}_{\Sigma}=\frac{\left(\hat{p}_{i}+A^{z}_{i}\sigma_{z}\right)^{2}}{2m_{i}^{*}}+\frac{\left(A^{\|}_{i}\right)^{2}}{2m_{i}}+v\left({\bf
A}_{y}^{\|}\cdot{\bm{\sigma}}\right).$ (4)

Hamiltonian (4) is gauge-invariant: as per Eq. (2), the $A^{z}_{i}$ in the
kinetic energy transforms as electromagnetic vector potential, while ${\bf
A}_{i}^{\|}=(A_{i}^{x},A_{i}^{y})$ in the remaining terms transforms by
rotation around $\hat{z}$. Moreover, the terms above are the only ones allowed
by symmetry to second order in momentum, keeping in mind symmetry under
reflection $x\rightarrow-x$ footnote:x-reflection . The astute reader will
also notice that the second term in Hamiltonian (4) is small relative to the
third one as long as the characteristic length scale of the texture remains
large compared with the lattice spacing, that is as long as the continuum
description of the texture applies. The concrete problem we treat in Section
III confirms this observation.

The non-uniformity of a texture appears in Hamiltonian (4) via the vector
potential $({\bf A}\cdot{\bm{\sigma}})$ that couples the electron spin to its
orbital motion. However, obeying different gauge transformation rules as per
Eq. (2), the $z$-component $A^{z}_{i}\sigma_{z}$ and the ‘in-plane’ spin
components $({\bf A}_{y}^{\|}\cdot{\bm{\sigma}})$ produce spin-orbit coupling
differently: The former induces ‘gauge’ spin-orbit coupling via the Peierls
substitution in the kinetic energy, while $v\left({\bf
A}_{y}^{\|}\cdot{\bm{\sigma}}\right)$ acts as a texture-induced Zeeman field.
In Section IV, we will discuss the peculiarities of this spin-orbit coupling
in relation to the work Pekar-1964  of Pekar and Rashba.

Finally, note that $m_{y}^{*}$ in Eq. (4) is renormalized relative to $m_{y}$
in the expansion of $\varepsilon_{{\bf p}_{0}}(\hat{\bf p})$ and
$\varepsilon_{{\bf p}_{0}+{\bf Q}}(\hat{\bf p})$, as per
$(m_{y}^{*})^{-1}=(m_{y})^{-1}+\frac{v^{2}}{\Delta}$. Therefore, $m_{y}^{*}$
is small against $m_{y}$:
$\frac{m_{y}^{*}}{m_{y}}\sim\frac{\Delta}{\epsilon_{F}}\ll 1$, while
$m_{x}^{*}=m_{x}$ is of the order of band electron mass $m$ or greater
footnote:m_x . Such a mass anisotropy arises for any (not only Néel)
$(\frac{\pi}{a},\frac{\pi}{a})$ order with a gap $\Delta\ll\epsilon_{F}\equiv
m_{y}v^{2}$. Electron-doped cuprates at low-to-optimal doping provide a
prominent example Armitage ; Matsui ; Song , even if the nature of their
ordering remains controversial, with experimental evidence presented both
against luke90 ; moto07 ; mang04a ; saad15  and for yama03 ; kang05 ; daga05 ;
yu07 ; dora18 ; RR-2021  the presence of (quasi)static Néel order.

##  III A tractable example: Belavin-Polyakov Skyrmion

Having obtained Hamiltonian (4), let us turn to a concrete problem: the $({\bf
A}\cdot{\bm{\sigma}})$ defined by a single Skyrmion. Consider a
centrosymmetric isotropic antiferromagnet with stiffness $J$ and continuum-
limit energy density $J({\bm{\nabla}}\hat{\bf n}_{\bf r})^{2}$. In the
topological sector with winding number $\mathcal{Q}=0,\pm 1,\pm 2...$, the
lowest-energy solution is the Belavin-Polyakov (BP) Skyrmion Belavin-1975 ;
Rajaraman  defined by a single length scale: the radius $R$. The energy $4\pi
J|\mathcal{Q}|$ of the BP Skyrmion is independent of $R$ by virtue of scale
invariance of energy $J\int d^{2}{\bf r}({\bm{\nabla}}\hat{\bf n}_{\bf
r})^{2}$. Being defined by a single length scale makes the BP Skyrmion the
simplest case to analyze, which leads us to study Hamiltonian (4) for a
$\mathcal{Q}=1$ BP Skyrmion. We focus on a configuration $\hat{\bf n}_{\bf
r}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ with the polar angle
$\theta$ depending only on the distance $r=\sqrt{x^{2}+y^{2}}$ to the Skyrmion
center, and $\phi=\arctan\frac{y}{x}$ being the azimuthal angle in the $(x,y)$
plane. The energy density $J({\bm{\nabla}}\hat{\bf n}_{\bf r})^{2}$ of such a
configuration is invariant under shifting $\phi$ by a constant $\gamma$,
usually called ‘helicity’ Goebel-2021 . Moreover, for the BP Skyrmion, the
helicity drops out of the electron problem up to the direction of the
resulting spin polarization. This allows us to reduce the setting to the
$\gamma=0$ pattern above, commonly called the ‘Néel’ Skyrmion.

Note that the latter has its own localized eigenexcitations Kravchuk-2019 ,
which, generally, shall be treated on an equal footing with the electron
degrees of freedom. However, for a sufficiently small single-ion anisotropy
and a not too large BP Skyrmion, the proper Skyrmion frequencies are small
against those of electron motion (see Appendix A). In this limit, treating the
electron problem as if the Skyrmion were perfectly static is a reasonable
first approximation that we now focus on.

To proceed, we need to fix the gauge, that is to select a concrete $U_{\bf
r}$. We do so by choosing

$U_{\bf r}=({\bf m}_{\bf r}\cdot{\bm{\sigma}}),$ (5)

with unit vector ${\bf m}_{\bf r}$ pointing along the bisector between
$\hat{z}$ and ${\bm{\Delta}}_{\bf r}$. Being equivalent to $\pi$-rotation
around ${\bf m}_{\bf r}$, such a $U_{\bf r}$ brings ${\bm{\Delta}}_{\bf r}$ to
point along $\hat{z}$ Tatara-2008 .

The $\mathcal{Q}=1$ BP Skyrmion profile is $\sin\theta=\frac{2z}{1+z^{2}}$
with $z=\frac{r}{R}$ Belavin-1975 ; Rajaraman  (in passing, notice that
$\theta\left[R\right]=\frac{\pi}{2}$). In the chosen gauge, calculation of
spin-$z$ components $A^{z}_{i}$ in the first term of Eq. (4) yields

$A^{z}_{x}=\frac{-\hbar y}{R^{2}+r^{2}}\,\,,\,\,A^{z}_{y}=\frac{\hbar
x}{R^{2}+r^{2}}.$ (6)

Thus, the Skyrmion produces geometric flux $\pm 2\pi\hbar$ for the spin-up and
spin-down components of the wave function: $\oint A^{z}_{i}dl_{i}=2\pi\hbar$,
with the integral taken along a large contour of radius $r\gg R$ footnote:flux
. Such a flux induces topological spin Hall effect Yin-2015 ; Buhl-2017 ;
Akosa-2018 .

The last term in Hamiltonian (4) takes the form

$v\left({\bf A}_{y}^{\|}\cdot{\bm{\sigma}}\right)=-\frac{\hbar
v}{R}\frac{\sigma_{x}}{1+z^{2}}=-\Delta\frac{\xi}{R}\frac{\sigma_{x}}{1+z^{2}},$
(7)

where, by analogy with superconductivity, we choose to call $\xi=\hbar
v/\Delta$ the antiferromagnetic coherence length. This term couples the
electron spin to its orbital motion and produces an attractive potential for
the spin-up component of the wave function along the $\hat{x}$ axis
footnote:BP .

Finally, the second term in Eq. (4) creates a repulsive potential

$\frac{\left(A^{\|}_{i}\right)^{2}}{2m_{i}}=\frac{\hbar^{2}}{2R^{2}}\left[\frac{1}{m_{x}}+\frac{1}{m_{y}}\right]\frac{1}{(1+z^{2})^{2}}.$
(8)

Comparing the terms (8) and (7), we see that the latter is an $a/R\ll 1$
fraction of the former. That is, the repulsion (8) is negligible relative to
the spin-orbit term (7) as long as the continuum description of the Skyrmion
is valid.

The term (7) is precisely the ‘texture-induced Zeeman’ part of the spin-orbit
coupling that we discussed below Eq. (4). Direct inspection shows that, for a
large (${R\gg\xi}$) Skyrmion, the r.h.s. of Eq. (7) overwhelms all the other
terms with $A^{\alpha}_{i}$ in Eq. (4) footnote:small , and creates non-
degenerate Skyrmion-electron bound states, a key result of our work. At
$R\gg\xi$, the low-lying bound states are shallow and spin-polarized in each
of the four $\Sigma$ valleys as shown in Fig. 1 footnote:chirality .

Note that this polarization is opposite to the one expected from Eq. (7). This
is a result of undoing the transformation $U_{\bf r}$ of Eq. (5) to restore
the original spin axes. While $U_{\bf r}$ is substantially non-uniform, the
reader will see that, for $R\gg\xi$, it does remain nearly constant over the
spatial extent of the low-lying bound states: $U_{\bf r}=({\bf m}_{\bf
r}\cdot{\bm{\sigma}})\approx\sigma_{z}$. That is, for low-lying bound states,
undoing the $U_{\bf r}$ amounts to a spin rotation by $\pi$ around $\hat{z}$,
which simply inverts the spin polarization relative to the one dictated by Eq.
(7). Notice that for smaller Skyrmions low-lying bound states are no longer
uniformly spin-polarized: instead, their spinor structure varies substantially
over the wave function range.

Figure 2:  Energy $\epsilon_{0}^{c}(R)$ of BP Skyrmion-electron bound state,
generated by the conduction band, and of its valence-band counterpart
$\epsilon_{0}^{v}(R)$ (the highest filled state), sketched as a function of
the BP Skyrmion radius $R$. In region I ($R\gg\xi=\frac{\hbar v}{\Delta}$),
the bound state is described by low-energy Hamiltonian (4). The
$\epsilon_{0}^{c}(R)$ is given by Eq. (9) and shown by solid line. In region
II ($\bar{R}\ll R\lesssim\xi$), the low-energy approximation breaks down, and
the bound state (dashed line) must be found from the full Hamiltonian of a
kind (1) in the presence of the Skyrmion, which goes beyond the scope of this
work. In region III (narrow range ${R-\bar{R}\ll\bar{R}}$), the bound state,
shown by solid line, becomes shallow again. Its disappearance at $R=\bar{R}$
can be described by low-energy Hamiltonian (4) (see main text). Pale lines
sketch the higher bound states.

The mass anisotropy
$\frac{m_{y}^{*}}{m_{x}^{*}}\sim\frac{\Delta}{\epsilon_{F}}\ll 1$ of
Hamiltonian (4) makes the $y$ coordinate ‘fast’ relative to $x$, and the
energies of low-lying bound states can be readily evaluated in the Born-
Oppenheimer approximation Tully . For $R\gg\xi$, the energies
$\epsilon_{n}^{c}$ of the bound states generated by the conduction band and
labelled by quantum number $n$ can be evaluated by expanding the r.h.s. of Eq.
(7) to first order in $z^{2}$ and finding the spectrum of the ensuing harmonic
oscillator with respect to $y$:

$\epsilon_{n}^{c}(R)\approx-\Delta\frac{\xi}{R}\left[1-\sqrt{\frac{2\xi}{R}}\left(n+\frac{1}{2}\right)\right].$
(9)

Subsequent account of the ‘slow’ coordinate $x$ generates energy levels that
we label with quantum number $k$, with the effective oscillator frequency of
the order of ${\sqrt{\frac{m_{y}}{m_{x}}\frac{a}{\xi}}\ll 1}$ relative to the
one above:

$\epsilon_{nk}^{c}(R)\approx-\Delta\frac{\xi}{R}\left[1-\sqrt{\frac{2\xi}{R}}\left\\{\left(n+\frac{1}{2}\right)+\sqrt{\frac{m_{y}}{m_{x}}\frac{a}{\xi}}\left(k+\frac{1}{2}\right)\right\\}\right].$
(10)

Needless to say, this ‘fine’ structure is much more prone to effects of proper
eigenexcitations Kravchuk-2019  of the Skyrmion than the r.h.s. of Eq. (9),
see Appendix A.

For $R\gg\xi$, the bound states (9-10) are shallow
(${|\epsilon_{0}^{c}|\ll\Delta}$), and the low-energy approximation of
Hamiltonians (3) and (4) remains valid. However, with $R$ decreasing,
$|\epsilon_{0}^{c}|$ grows to attain the order of $\Delta$ at $R\sim\xi$,
where the low-energy approximation breaks down along with Hamiltonians (3) and
(4), as sketched in Fig. 2.

Now we will show that, with $R$ decreasing further below $\xi$, the bound
state becomes shallow again, and vanishes at an
$\bar{R}\sim\sqrt{\frac{m_{y}}{m_{x}}}\sqrt{\xi a}$. To make this length scale
manifest, we eliminate $A^{z}_{y}$ from the first term in Eq. (4) by gauge
transformation

$W=e^{i\sigma_{z}\chi},\,\,\chi\left(\tilde{x},\tilde{y}\right)=\frac{-\tilde{x}}{\sqrt{1+\tilde{x}^{2}}}\arctan{\frac{\tilde{y}}{\sqrt{1+\tilde{x}^{2}}}},$
(11)

where $\tilde{x}=\frac{x}{R}$ and $\tilde{y}=\frac{y}{R}$. As a result,
Hamiltonian (4) takes the form

$\tilde{\mathcal{H}}_{\Sigma}=\frac{\hat{p}_{y}^{2}}{2m_{y}^{*}}+v\left(\tilde{\bf
A}_{y}^{\|}\cdot{\bm{\sigma}}\right)+\frac{\left(A^{\|}_{i}\right)^{2}}{2m_{i}}+\frac{\left(\hat{p}_{x}+\tilde{A}^{z}_{x}\sigma_{z}\right)^{2}}{2m_{x}},$
(12)

where $\tilde{A}^{z}_{x}=A^{z}_{x}+\partial_{x}\chi$, and

$(\tilde{\bf A}_{y}^{\|}\cdot{\bm{\sigma}})=A_{y}^{\|}\left[\sigma_{x}\cos
2\chi+\sigma_{y}\sin 2\chi\right].$

In terms of the ‘fast’ coordinate $y$, Hamiltonian (12) describes a particle
in a one-dimensional potential $\mathcal{U}_{x}(y)$, parametrically dependent
on the ‘slow’ variable $x$, which again invites the Born-Oppenheimer
approximation. Comparing the characteristic value $\frac{\hbar
v}{R}=\Delta\frac{\xi}{R}$ of the second term in Eq. (12) with typical kinetic
energy
${\frac{\hbar^{2}}{m_{y}^{*}R^{2}}\approx\Delta\left(\frac{\xi}{R}\right)^{2}}$
of the trapped electron, we see that the former is indeed small against the
latter for $R\ll\xi$, the range in question. The effective bound-state energy
is thus defined by the integrated potential ${u(x)=\int\mathcal{U}_{x}(y)dy}$
LL-III . Contribution of the second term in Eq. (12) to $u(x)$ is

$u_{1}(x)\sigma_{x}=-v\int dy\left(\tilde{\bf
A}_{y}^{\|}\cdot{\bm{\sigma}}\right)=\hbar
v\sigma_{x}\frac{\sin\left(\frac{\pi\tilde{x}}{\sqrt{1+\tilde{x}^{2}}}\right)}{\tilde{x}}.$
(13)

The third term in Eq. (12) is an $\frac{a}{R}\ll 1$ fraction of the second,
thus its contribution to $u(x)$ can be neglected as long as continuum
description of the Skyrmion applies ($R\gg a$). By contrast,

$u_{2}(x)=\frac{1}{2m_{x}}\int dy(\tilde{A}^{z}_{x})^{2},$ (14)

arising from the last term, requires care since $(\tilde{A}^{z}_{x})^{2}$
remains finite as $y\rightarrow\infty$:

$A^{2}(x)\equiv\lim_{y\rightarrow\infty}(\tilde{A}^{z}_{x})^{2}=\frac{\hbar^{2}}{R^{2}}\frac{[\pi/2]^{2}}{(1+\tilde{x}^{2})^{3}},$

which makes $u_{2}(x)$ diverge. We remedy this by writing
$(\tilde{A}^{z}_{x})^{2}=\left[(\tilde{A}^{z}_{x})^{2}-A^{2}(x)\right]+A^{2}(x)$.
The term in the square brackets then gives a finite contribution to $u(x)$,
which is suppressed relative to $u_{1}(x)$ by a factor
$\frac{m_{y}}{m_{x}}\frac{a}{R}$, and hence remains negligible within
continuum description ($R\gg a$). The resulting bound-state energy $w_{-}(x)$
at a given $x$ is thus defined LL-III  by $u_{1}(x)$ alone:

$w_{-}(x)=-\frac{m_{y}^{*}}{2\hbar^{2}}\left[u_{1}(x)\right]^{2}=-\frac{\Delta}{2}\frac{\sin^{2}\left(\frac{\pi\tilde{x}}{\sqrt{1+\tilde{x}^{2}}}\right)}{\tilde{x}^{2}}.$
(15)

It competes with the repulsive contribution of $A^{2}(x)$:

$w_{+}(x)=\frac{A^{2}(x)}{2m_{x}}=\frac{\hbar^{2}}{2m_{x}R^{2}}\frac{[\pi/2]^{2}}{(1+\tilde{x}^{2})^{3}}.$
(16)

As per Eqs. (6) and (11), $\tilde{A}^{z}_{x}$ is odd with respect to $y$, thus
the cross-term
$\left\\{\hat{p}_{x},\tilde{A}^{z}_{x}\sigma_{z}\right\\}/2m_{x}$ averages out
upon integration over $y$. As a result, upon switching to the dimensionless
coordinate $\tilde{x}=\frac{x}{R}$, the Hamiltonian $\tilde{\mathcal{H}}_{x}$
reads

$\tilde{\mathcal{H}}_{x}=\frac{\hbar^{2}}{2m_{x}R^{2}}\left[-\frac{d^{2}}{d\tilde{x}^{2}}+\frac{[\pi/2]^{2}}{(1+\tilde{x}^{2})^{3}}\right]-\frac{\Delta}{2}\frac{\sin^{2}\left[\frac{\pi\tilde{x}}{\sqrt{1+\tilde{x}^{2}}}\right]}{\tilde{x}^{2}}.$
(17)

Taken alone, the last term above is obviously beyond the low-energy
approximation. However, with decreasing $R$, the repulsion grows relative to
attraction, and overcomes it at an
$\bar{R}\sim\sqrt{\frac{m_{y}}{m_{x}}}\sqrt{\xi a}\ll\xi$. Thus, Hamiltonian
(17) is valid only in a narrow range $R-\bar{R}\ll\bar{R}$, where the bound
state becomes shallow to disappear at $R=\bar{R}$. Notice that $\bar{R}\gg a$
as long as the ratio $\frac{m_{y}}{m_{x}}$ is not too small
($\frac{m_{y}}{m_{x}}\gg\frac{\Delta}{\epsilon_{F}}$).

Note that neither the bound state becoming shallow in a narrow range near
$\bar{R}$ nor its disappearance at $\bar{R}$ rely on the mass anisotropy: the
same behavior obtains for a perfectly isotropic mass, where the Hamiltonian
can be diagonalized by solving a single equation for the radial wave function.

Described in this Section, the bound-state behavior at large and small
Skyrmion radii has several implications. These are convenient to illustrate at
half-filling, where the total number of states (itinerant plus bound)
generated by the valence band in the presence of the Skyrmion equals the
number of electrons – that is, the number of unit cells in the sample. By
particle-hole symmetry, every bound state $\epsilon_{\alpha}^{c}(R)$, split
off the conduction band, has a valence-band counterpart
$\epsilon_{\alpha}^{v}(R)=-\epsilon_{\alpha}^{c}(R)$, as shown in Fig. 2.
Therefore, for any $R$, the number of negative-energy states equals the number
of electrons. Hence, at zero temperature, all the positive-energy states are
empty while all the negative-energy states are filled. Last but not the least,
notice that the negative-energy bound states rise above the top of the valence
band, thus increasing the energy cost of the Skyrmion. Qualitatively, the
energy $\epsilon_{0}^{v}(R)$ of the highest filled state behaves as sketched
in Fig. 2.

##  IV Discussion and conclusions

The Skyrmion-electron bound states of Section III owe their existence to the
‘texture-induced Zeeman’ part of the spin-orbit coupling, the last term of
Hamiltonian (4). Unlike the ‘gauge’ term, arising from the Peierls
substitution in the kinetic energy Volovik-1987 ; Tatara-2008 , this term has
no equivalent in a ferromagnet. At the same time, it hinges on the lower
symmetry of $\Sigma$ points in the Brillouin zone: at the corner points $X$ or
at the center point $\Gamma$ in Fig. 1, such a term is not allowed.

Notice that, in the effective-mass Hamiltonian of Pekar and Rashba, the spin-
orbit coupling is linear in momentum (see Eqs. (4) and (5) of Ref. Pekar-1964
), as is the ‘gauge’ spin-orbit term in Hamiltonian (4). By contrast, the
‘texture-induced Zeeman’ spin-orbit coupling does not involve electron
momentum, and hence is qualitatively different.

It is instructive to compare the latter term with other known spin-orbit
couplings. To gain perspective, recall that the textbook Pauli spin-orbit
coupling appears in the Schrödinger Hamiltonian only as a relic of relativity,
in second order of the expansion in the inverse speed of light $1/c$ LL-IV .
Remarkably, in antiferromagnets subject to magnetic field, spin-orbit coupling
may appear in first order in $1/c$, via a substantial momentum dependence of
the $g$-tensor RR-2021 ; RR-2008 ; RR-2009 . In contrast to all of the above,
the texture-induced spin-orbit coupling of Eq. (4) does not involve $1/c$, or
the fine-structure constant $\alpha=\frac{e^{2}}{\hbar c}\approx 1/137$, at
all. Instead, the energy scale of the ‘texture-induced Zeeman’ spin-orbit term
${v({\bf A}^{\|}_{y}\cdot{\bm{\sigma}})\sim\hbar v/L}$ is defined by the
relevant length scale $L$ of the texture and by the Fermi velocity ${v\sim
Wa/\hbar}$, with $W$ the electron bandwidth and $a$ the lattice spacing. For
the Belavin-Polyakov Skyrmion we analyzed in Section III, $L$ is given by the
Skyrmion radius $R$. Compared with $W$, the ‘texture-induced Zeeman’ term is
small only in the measure of $L$ being large against $a$: ${v({\bf
A}^{\|}_{y}\cdot{\bm{\sigma}})\sim W\frac{a}{L}}$ (recall that continuum
description is limited to $L\gg a$). Relative to the gap $\Delta$, the term is
small in the measure of $L$ being large against the Néel coherence length
$\xi=\frac{\hbar v}{\Delta}$, that is ${v({\bf
A}^{\|}_{y}\cdot{\bm{\sigma}})\sim\Delta\frac{\xi}{L}}$.

The Skyrmion-electron bound states are nondegenerate. Being limited by the
gap, their energy scale is given by $\Delta$. In this regard, the texture-
induced spin-orbit coupling in Eq. (4) is a real-space analogue of large band-
splitting effects, discussed for certain types of antiferromagnets Pekar-1964
; Hayami-2019 ; Yuan-2020 ; Reichlova-2020 ; Smejkal-2022 .

We have shown that, in a Néel antiferromagnet with certain locations of the
electron band extrema, a Skyrmion produces non-degenerate electron bound
states. In each $\Sigma$-valley, low-lying bound states are spin-polarized as
shown in Fig. 1. By virtue of charge neutrality, at half-filling the bound
states do not produce a charge density modulation. However, doping the half-
filled antiferromagnetic insulator by an extra carrier turns the Skyrmion into
a charged particle. This effect does not rely on the BP profile we used as an
illustration, and appears for any credible shape such as that of a domain-wall
Skyrmion.

Finally, a brief comment on how disorder may limit the validity of our
results. A crude bound becomes evident upon comparing the energy $\hbar v/R$
of the low-lying bound states with the disorder-induced scattering rate
$1/\tau$ of an electron: for a BP Skyrmion of radius $R\gg\xi$, the leading
term in Eq. (9) is valid as long as the electron mean free path $l=v\tau$ is
large compared with $R$.

Pekar and Rashba showed Pekar-1964  how a non-uniformity of magnetization
couples electron spin to its orbital motion. However, the analysis of the
preceding Sections involved no magnetization at all – not even locally. The
only quantity present was the staggered (${\bf
Q}=(\frac{\pi}{a},\frac{\pi}{a})$) magnetization and its inhomogeneity. Thus,
Pekar’s and Rashba’s insight holds beyond their original statement: in a
general magnetically ordered system, inhomogeneity begets spin-orbit coupling.

Skyrmion-electron bound states are a new arrival in the family of electron
states, localized on topological defects such as dislocations Landauer-1954 ,
vortices in superconductors Caroli-1964  or solitons in organic materials
Brakir-1984 ; Heeger-1988 . Becoming charged in the presence of dopant
carriers, the Skyrmion can be manipulated by electric field, which may open
new possibilities for its use in devices. We hope that our results stimulate
further work both on fundamental and applied aspects of this phenomenon.

###### Acknowledgements.

We thank P. Pujol for the many discussions and helpful suggestions. We are
grateful to Ya. B. Bazaliy, M. V. Kartsovnik and A. Monin for illuminating
comments. Finally, we thank G. Baskaran for pointing us, after this work
appeared on the arXiv, to early studies John-Golubentsev-1993 ; John-
Golubentsev-1995  that found Skyrmion-electron bound states in an exotic spin
liquid (see Appendix B).

##  Appendix A When can one treat the Belavin-Polyakov Skyrmion as static?

For $R\gtrsim\xi$, treating the BP Skyrmion profile as static input to the
electron problem requires the bound-state level spacing
$\hbar\omega_{e}\sim\Delta\left(\frac{\xi}{R}\right)^{\frac{3}{2}}$ in Eq. (9)
to be large against the energies of high-frequency localized eigenmodes of the
Skyrmion. These are bound from above by the spin wave gap $\hbar\Omega_{0}$ of
the bulk magnon spectrum Kravchuk-2019 . In the relevant limit of small
single-ion anisotropy $K\ll J$, the behavior $\hbar\Omega_{0}\sim\sqrt{KJ}$
Kittel-1951 ; Rezende  translates the condition $\omega_{e}\gg\Omega_{0}$ into
$\Delta\left(\frac{\xi}{R}\right)^{3/2}\gg\sqrt{KJ}$. Which means that a
sufficiently large BP Skyrmion can no longer be treated as static. For a weak
enough anisotropy, this limits our treatment to radia

$R\ll\xi\left(\frac{\Delta}{J}\right)^{2/3}\left(\frac{J}{K}\right)^{1/3}.$
(18)

Inequality (18) is meaningful only if its r.h.s. is large compared with $\xi$,
that is if

$\frac{K}{J}\ll\left(\frac{\Delta}{J}\right)^{2}.$ (19)

Now, the conditions (18), (19) define the possibility of treating the Skyrmion
as static when dealing with $y$, the ‘fast’ coordinate of the electron. As per
Eq. (10), for the ‘slow’ coordinate, $x$, the relevant frequency is
$\hbar\omega_{e}^{x}\sim\Delta\sqrt{\frac{m_{y}}{m_{x}}}\sqrt{\frac{a}{\xi}}\left(\frac{\xi}{R}\right)^{\frac{3}{2}}\ll\hbar\omega_{e}$.
As a result, the condition for treating the Skyrmion as static when dealing
with $x$ is more stringent than (18):

$R\ll\xi\left(\frac{\Delta}{\epsilon_{F}}\right)^{1/3}\left(\frac{m_{y}}{m_{x}}\right)^{1/3}\left(\frac{\Delta}{J}\right)^{2/3}\left(\frac{J}{K}\right)^{1/3}.$
(20)

Inequality (20) makes sense only if its r.h.s. is large compared with $\xi$,
that is if

$\frac{K}{J}\ll\left(\frac{\Delta}{J}\right)^{2}\frac{m_{y}}{m_{x}}\frac{\Delta}{\epsilon_{F}}.$
(21)

Put otherwise, the levels emerging from quantizing the electron motion along
the ‘slow’ coordinate $x$ define the ‘fine’ structure of the bound-state
spectrum as opposed to the ‘gross’ structure arising from quantization along
the ‘fast’ coordinate $y$. This ‘fine’ structure can be treated in the static-
Skyrmion approximation under conditions (20), (21) that are, naturally, much
more stringent than similar inequalities (18), (19) for the ‘gross’ structure
of Eq. (9). At the same time, no matter how small the $\omega_{e}/\Omega_{0}$
ratio, a sufficiently large number of Skyrmion eigen-excitations would
eventually influence the excited electron bound states (at an appropriately
high order of perturbation theory).

##  Appendix B Earlier work John-Golubentsev-1993 ; John-Golubentsev-1995 ;
Cheng-2012 ; Haas-1996 ; Morinari-2012 ; Shraiman-1988

In this section of the Appendix, we discuss some of the relevant early work.

S. John and A. Golubentsev John-Golubentsev-1993 ; John-Golubentsev-1995
studied the Hubbard model on a square lattice in a topological spin liquid
state, defined by two key properties: (i) checkerboard Néel order, and (ii)
anti-periodicity of the electron wave function along closed path around any
elementary plaquette of the lattice. As a consequence, the unit cell in such a
state quadruples relative to the underlying square lattice, and the electron
wave function is thus a 4-spinor. The model band extrema fall at the very same
points $\Sigma=(\pm\frac{\pi}{a},\pm\frac{\pi}{a})$ that we focussed on, and
all the four points $\Sigma$ are equivalent in the ‘ordered’ Brillouin zone.
The resulting ‘Dirac’ electron spectrum near $\Sigma$ is thus isotropic by
symmetry.

As opposed to the above, in the Néel state of our interest the unit cell
doubles rather than quadruples and, therefore, the electron wave function is a
bispinor rather than a 4-spinor. Contrary to being all equivalent in the spin
liquid state, in the Néel phase the four points $\Sigma$ split into two
inequivalent pairs $\Sigma_{1,2}$ and $\Sigma_{3,4}$, shown in Fig. 1. In
contrast to the spin-liquid state, the symmetry of the Néel state does not
require the electron spectrum near the $\Sigma$ points to be isotropic. Quite
to the contrary, it tends to be strongly anisotropic, consistently with
experimental findings Armitage ; Matsui ; Song  in a number of cuprates.

Lastly, the Skyrmion-electron bound states we found are non-degenerate. In
each of the four $\Sigma$-valleys, low-lying bound states are spin-polarized
as shown in Fig. 1, whereas the bound states of Refs. John-Golubentsev-1993 ;
John-Golubentsev-1995  are doubly degenerate as a consequence of the elevated
symmetry of the spin-liquid state.

To summarize, the similarities between the Skyrmion-electron bound states of
our work and those found by John and Golubentsev arise from Néel order being
an ingredient of the spin liquid studied in Refs. John-Golubentsev-1993 ;
John-Golubentsev-1995 . Such similarities include the shallow character of
bound states at large BP Skyrmion radia, although John and Golubentsev have
not explored the low-energy limit, instead opting for solving their full
8$\times$8 Hamiltonian numerically.

The differences between our results and those of John and Golubentsev stem
from the features of their spin liquid that are not inherent to a generic Néel
state that we studied, such as (i) an elevated symmetry of the spin liquid,
and (ii) the wave function antiperiodicity under translation around any
elementary plaquette of the square lattice. Such differences lead to double
degeneracy of the bound states found by John and Golubentsev, while ours are
non-degenerate.

Now we turn to the work Cheng-2012  by R. Cheng and Q. Niu, who derived
coupled quasiclassical equations of motion for the electron momentum,
coordinate and spin in an antiferromagnetic texture. By construction, such a
description holds only for large quantum numbers and, in fact, the equations
of Cheng and Niu become singular at the MBZ boundary ($\xi\rightarrow 0$), see
the r.h.s. of Eq. (8c) of Ref. Cheng-2012 . By contrast, our effective
electron Hamiltonian (3-4) covers both the quasiclassical regime and the
extreme quantum limit as long as the electron energies are close to the gap
edge ($\frac{|E-\Delta|}{\Delta}\ll 1$).

Finally, we would like to mention the work Haas-1996 ; Morinari-2012  on
different versions of the $t$-$J$ model of high-temperature superconductivity.
These studies were performed on small clusters in the atomic limit, with
exchange integrals substantial or even large compared with the hopping matrix
elements. The authors of both Refs. Haas-1996 ; Morinari-2012  reach a
conclusion that is, in a way, reciprocal to ours (“a Skyrmion produces
electron bound states”): namely, that introducing a dopant carrier into a
single CuO plane renders a Skyrmion Haas-1996  or a half-Skyrmion
Morinari-2012  configuration energetically favorable Shraiman-1988 . When
applied to a single dopant carrier bound to a Skyrmion, this conclusion stems
from neglecting the valence band giving rise to filled bound states, whose
energy increases rather than decreases (see Fig. 2), and thus overwhelms the
energy gain found in Refs. Haas-1996 ; Morinari-2012 . A more technical
difference with respect to our work is that the studies Haas-1996 ;
Morinari-2012  were performed in the atomic limit, and thus produced Skyrmion
sizes of the order of the lattice spacing, where the continuum description is
not applicable.

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The Size-Ramsey Number of 3-uniform Tight Paths

jie]Jie HanPartially supported by the Simons Collaboration Grant for Mathematicians #630884.
yoshi]Yoshiharu KohayakawaPartially supported by CNPq (311412/2018-1, 423833/2018-9) and FAPESP (2018/04876-1, 2019/13364-7)
shoham]Shoham LetzterResearch supported by the Royal Society.
gui]Guilherme Oliveira MotaPartially supported by CNPq (304733/2017-2, 428385/2018-4) and FAPESP (2018/04876-1, 2019/13364-7).
olaf]Olaf ParczykPartially supported by Technische Universität Ilmenau, the Carl Zeiss Foundation, and the DFG (Grant PA 3513/1-1).The cooperation of the authors was supported by a joint CAPES/DAAD
PROBRAL project (Proj. 430/15, 57350402, 57391197).

This study was financed in part by CAPES, Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior, Brazil, Finance
Code 001.

FAPESP is the São Paulo Research Foundation.  CNPq is the National
Council for Scientific and Technological Development of

Given a hypergraph $H$, the size-Ramsey number $\hat{r}_2(H)$ is the smallest
integer $m$ such that there exists a hypergraph $G$ with $m$ edges with the property
that in any colouring of the edges of $G$ with two colours there is a
monochromatic copy of $H$.
We prove that the size-Ramsey number of the
$3$-uniform tight path on $n$ vertices $P^{(3)}_n$ is linear in $n$,
i.e., $\hat{r}_2(P^{(3)}_n) = O(n)$.
This answers a question by Dudek, La Fleur, Mubayi, and Rödl for
$3$-uniform hypergraphs [On the
size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016),
417–434], who proved $\hat{r}_2(P^{(3)}_n) = O(n^{3/2} \log^{3/2} n)$.

§ INTRODUCTION

For hypergraphs $G$ and $H$ and an integer $s$, we denote by $G \rightarrow
(H)_s$ the property that in any $s$-colouring of the edges of $G$ there is a
monochromatic copy of $H$.
The $s$-colour size-Ramsey number $\hat{r}_s(H)$ is

\begin{align*}
\sr_s(H) := \min \{ |E(G)| : G \rightarrow (H)_s \}.
\end{align*}

For the $n$-vertex path $P_n$, Erdős [11] asked if $\sr_2(P_n)=O(n)$, which was
answered positively by Beck [3] using the probabilistic method.
An explicit construction for the same results was given by Alon and
Chung [1].
Many successive improvements led to the currently best known bounds $3.75n -
o(n)\leq\sr_2(P_n)\leq 74n$ (see, e.g.,
[3, 6, 10, 2] for lower
bounds, and [3, 9, 14, 10] for
upper bounds).
For $s\geq 2$ colours, Dudek and Prałat [10] and
Krivelevich [13] proved that there are constants $c$ and
$C$ such that $cs^2n\leq\sr_s(P_n)\leq Cs^2(\log s)n$.

The systematic investigation of size-Ramsey questions for hypergraphs was
initiated by Dudek, La Fleur, Mubayi, and Rödl [8].
Besides cliques and trees, they studied generalisations of paths.

We say that an $r$-uniform hypergraph is an $\ell$-path if
there exists an ordering of its vertices such that every edge is
composed of $r$ consecutive vertices, two (vertex-wise) consecutive
edges share exactly $\ell$ vertices, and every vertex is contained in
an edge.
For $1 \le \ell \le r-1$, let $P^{(r)}_{n,\ell}$ denote the $r$-uniform $\ell$-path on $n$
vertices and for the tight path, where $\ell=r-1$, we write $P^{(r)}_{n}$.
Dudek, La Fleur, Mubayi, and Rödl [8] deduced from Beck's result [3] that $\hat{r}_2(P^{(r)}_{n,\ell}) = O(n)$, when $1 \le
\ell \le r/2$.
Furthermore, they proved that $\hat{r}_2(P^{(r)}_{n})
= O_r(n^{r-1-\alpha} \log^{1+ \alpha}n)$ with $\alpha=(r-2)/(\binom{r-1}{2}+1)$, which
gives $\hat{r}_2(P^{(3)}_{n}) = O(n^{3/2} \log^{3/2}n)$.

This was improved and extended to more colours by Lu and
Wang [15], who showed that $\hat{r}_s(P^{(r)}_{n}) = O_r(s^r (n \log
n)^{r/2})$ for $s \ge 2$ colours.
Dudek, La Fleur, Mubayi, and Rödl [8] asked if $\hat{r}_2(P^{(r)}_n)=O_r(n)$ for $r \ge 3$.
We answer this question for $3$-uniform hypergraphs by proving the following result.

The $2$-colour size-Ramsey number of the $3$-uniform tight path is

\begin{align*}
\sr_2(P^{(3)}_n) = O(n).
\end{align*}

Trivially, we need at least $n$ edges, so this is asymptotically optimal.
As observed in [8], bounds on size-Ramsey numbers
for some uniformity can be used to obtain bounds for larger uniformities.
We obtain the following corollary.

For any integer $r$ such that $3\mid r$, the $2$-colour size-Ramsey
number of the $r$-uniform $(2r/3)$-path is

\begin{align*}
\sr_2(P^{(r)}_{n,2r/3}) = O(n).
\end{align*}

To see this, take the graph given by
Theorem <ref> and replace every vertex by
a set of $r/3$ vertices. Then each $3$-edge naturally
gives an $r$-edge, and every $3$-uniform tight path
becomes an $r$-uniform $(2r/3)$-path.

Our proof combines new ideas and the method developed by Clemens, Jenssen, Kohayakawa,
Morrison, Mota, Reding, and Roberts [7] for estimating the
size-Ramsey number of powers of paths (see also [5, 12]).
It is plausible that ideas from [12, 5] may provide a strategy to solve the case with $s \ge 3$ colours.
However, the question whether the size-Ramsey number of a tight path is linear for hypergraphs with uniformity $r\ge 4$ remains open and requires additional ideas.

§ PRELIMINARIES

In this short section, we give a sketch of our proof of
Theorem <ref> and state two simple lemmas about random graphs.

§.§ Sketch of the proof of Theorem <ref>

We will first sketch a proof for $\sr_2(P_n) = O(n)$.  It is not hard
(cf. Lemmas <ref> and <ref> below) to obtain a
graph $G$ with $O(n)$ edges such that for any two sufficiently large
and disjoint sets of vertices $A$ and $B$ there is a path of
length $n$ alternating between $A$ and $B$.  Given such a graph $G$,
we show that
$G \rightarrow (P_n)_2$.  Consider an arbitrary $2$-colouring of the
edges of $G$ with colours blue and red.
If there is no blue $P_n$ in $G$ we can show
(cf. Lemma <ref> below) that there are two
sets $A$ and $B$ of size at least $n$ with no blue edges in between.
By the property of $G$ mentioned above there exists a $P_n$
alternating between $A$ and $B$, which unequivocally has to be red.

For the proof of Theorem <ref> we follow, in principle, the
same strategy.  Based on a blow-up of a power of a similar graph $G$,
we define a $3$-uniform hypergraph $H$ and claim that
$H \rightarrow (P^{(3)}_{n})_2$.
We define an auxiliary (generalised)
graph $F$ on $V(G)$, which has $2$- and $3$-edges, such that a long
path in $F$ gives a blue $P^{(3)}_n$ in $H$.
If $F$ does not contain a long path, then we find a family of disjoint
sets such that no edge of $F$ lies between these sets
(cf. Lemma <ref>).  Then by the properties of $G$
there exists a path in $G$ alternating through these sets.  As there
are no edges of $F$ `interfering' with this path, we are able to turn
it into a red $P^{(3)}_n$ in $H$.

In the next section we provide the lemmas needed to obtain $G$.
Afterwards, in Section <ref> we introduce the notion of
$(2,3)$-graphs, which, as can be seen above, plays a key role in
our argument.  Finally, we prove Theorem <ref> in
Section <ref>.

§.§ Sparse graphs with many long paths

The following two lemmas are proved in [7].
Basically, together they imply that for every $k$ and $n$ there exists
a graph $G$ with $O_k(n)$ edges such that, for any disjoint sets of
vertices $A_1,\dots,A_{k+1}$ that are large enough, there exists a
path of length $n$ `alternating' through these sets.

For every pair of positive constants $\varepsilon$ and $a$, there is a
constant $b$ such that, for any large enough $n$, there is a graph
$H$ with $v(H)=an$ and $\Delta(H)\leq b$ such that the following holds:

* For every pair of disjoint sets $S, T\subseteq V(H)$ with $|S|, |T|\geq\varepsilon n$, we get $e_H(S,T)>0$.

For every integer $k\geq 1$ and every $\eps>0$ there exists an
integer $a$ such that the following holds.  Let $H$ be a graph on at
least $am$ vertices such that for every pair of disjoint sets $S$,
$T\subseteq V(H)$ with $|S|$, $|T|\geq \eps m$ we have
$e_{H}(S,T)> 0$.  Then the following holds:

* For every family $A_1,\dots,A_{k+1}\subseteq V(H)$ of pairwise
disjoint sets each of size at least $\eps a m$, there is a path
$P_m=(x_1,\dots,x_m)$ in $H$ with $x_i \in A_j$ for
all $1\leq i\leq m$, where $j\equiv i\pmod{k+1}$.

Note that the hypothesis on $H$ in Lemma <ref>
is [prop:exist](P1$_m$) from Lemma <ref>.  Therefore, roughly
speaking, Lemma <ref> tells us that [prop:exist](P1$_m$)
implies <ref>.

§ (2,3)-GRAPHS

In this section we introduce a structure that helps us to transfer
some ideas from the graph case to the hypergraphs setting.  A
$(2,3)$-graph $F=(V,E)$ consists of a set of vertices $V$ and a
set $E$ of $2$-edges of the form $\{u,v\}$ and $3$-edges of
the form $(\{u,v\} ,w)$, for distinct vertices $u,v,w\in V$. For simplicity
we will write $uv$ for $\{u,v\}$ and $uv(w)$  for $(\{u,v\}, w)$.
A sequence of vertices $P =(x_1, \dots, x_m)$ is a $(2,3)$-path of length $m$ in $F$ if for
every $i=1,\dots,m-1$ either $x_ix_{i+1}\in E$ or
$x_ix_{i+1}(w_i)\in E$ for some $w_i\in V\setminus\{x_1,\dots,x_m\}$,
with all the $w_i$ distinct.

Given pairwise disjoint sets $V_1,\dots,V_{k+1}$, we say that an edge
$uv \in E(F)$ ($uv(w) \in E(F)$) is a transversal with respect to
$V_1,\dots,V_{k+1}$, if $u$ and $v$ ($u$, $v$, and $w$) are in different sets $V_i$.
When the sets $V_1,\dots,V_{k+1}$ are clear from the context we say that the edge is a

We want to prove that if a sufficiently large $(2,3)$-graph $F=(V,E)$ contains no $(2,3)$-path with $n$ vertices, then there exist large disjoint sets $V_1,\dots,V_{k} \subseteq V$  such that $E$ contains no transversals and that there is no edge $uv(w)$ with $u \in V_1 \cup \dots \cup V_{k-1}$ and $v,w \in V_{k}$.
The last property is only required to support our inductive proof.
To prove this we use a Depth First
Search (DFS) algorithm.  For example, Ben-Eliezer, Krivelevich, and
Sudakov [4] used a DFS algorithm to find long
paths in expanding graphs to obtain bounds on the size-Ramsey number
of directed paths.  Their algorithm traverses the vertices of the input
graph and maintains a set $S$ of vertices that are fully dealt with,
a set $U$ of currently active vertices, and a set $J$ of vertices that
were not considered so far.  The set $U$ always spans a path and in
every step, if at all possible, this path is extended by adding a vertex from $J$.
Otherwise, the last vertex of the path is removed and added to $S$.
It is immediate that there cannot be any edges between $S$ and $J$ and if $U$ stays small, then at
some point during the execution both $S$ and $J$ are large.

We adapt this algorithm to the setting of $(2,3)$-graphs (see Algorithm <ref> below).
As in the graph case we greedily extend a $(2,3)$-path (preferring $2$-edges over $3$-edges)
and backtrack if it gets stuck. We will now give the details of our algorithm. The input is a
$(2,3)$-graph $F=(V,E)$, disjoint subsets of vertices $V_1,\dots,V_k$, and an ordering of the
vertices $V=\{ v_1,\dots,v_N \}$.  During the algorithm we maintain
sets $S$, $T$, $W_S$, $W_U$, $T_i$ for $i\in[k]$ and a $(2,3)$-path $U$ as follows:

* $S \subseteq V'$ is the set of vertices that are fully dealt

* $W_S \subseteq V$ is the set of vertices $w$ that were `used' by
vertices from $S$.

* $U$ contains the currently active vertices in a $(2,3)$-path.

* $W_U \subseteq V$ is the set of vertices $w$ that are `used' by
the path $U$.

* $T_1\cup\dots\cup T_k$ are disjoint and $T_i\subset V_i$ for $i\in[k]$.

In every step of the algorithm, either the $(2,3)$-path $U$ is
extended by adding a vertex from $T_k$ to it or this is not possible,
and the last vertex from $U$ is removed and put into $S$.  While the
algorithm runs, after each execution of the while loop, we have the
following invariants, where $m$ is the length of the $(2,3)$-path

*  $U=(u_1,\dots,u_{m})$ is a $(2,3)$-path and
$W_U$ is the set of the vertices $w$ in the edges $u_iu_{i+1}(w)$
($1\leq i<m$) in the $(2,3)$-path $U$.

*  $S$, $U\subseteq V_k$, $W_U\subseteq T_1\cup\dots\cup T_{k-1}$, $T_i\subseteq V_i$ for $i\in[k]$, $|W_S| \le |S|$, and $|W_U| \le \max\{ 0, m-1 \}$.

This process is described in Algorithm <ref>.

A $(2,3)$-graph $F=(V,E)$, disjoint subsets of vertices $V_1,\dots,V_k$, and an ordering of the vertices $V=\{ v_1,\dots,v_N \}$.

Define $m \leftarrow 0$, $S\leftarrow \emptyset$, $W_S \leftarrow \emptyset$, $W_U \leftarrow \emptyset$, $T_i\leftarrow V_i$ for $1\leq i\leq k$
Let $v$ be the vertex with smallest index from $T_k$
$u_1 \leftarrow v$, $m\leftarrow 1$, $T_k \leftarrow T_k \setminus \{v\}$

Let $T_\ext \leftarrow \{ v\in T_k \colon u_mv \in E \text{ or } u_mv(w) \in E \text{ with }w \in T_1\cup\cdots\cup T_k \}$
$T_\ext\not= \emptyset$
Let $v $ be the vertex with the smallest index from $T_\ext$
$u_{m+1} \leftarrow v$, $T_k \leftarrow T_k \setminus \{v\}$
$u_mu_{m+1} \not \in E$

Let $w \in T_1\cup\cdots\cup T_k $ be the vertex of smallest index such that $u_mu_{m+1}(w) \in E$;
There is one because $u_{m+1} \in T_\ext$.
$W_U \leftarrow W_U \cup \{w\}$ and $T_i \leftarrow T_i \setminus \{w\}$, where $w\in T_i$

$m \leftarrow m+1$

$S \leftarrow S \cup \{ u_m \}$

$u_{m-1}u_m \not \in E$

Let $u_{m-1}u_m(w) \in E$ with $w \in W_U$ be the edge used by the $(2,3)$-path; This is well defined by <ref>.
$W_S \leftarrow W_S \cup \{w\}$ and $W_U \leftarrow W_U \setminus \{w\}$

$m \leftarrow m-1$

DFS algorithm for traversing a $(2,3)$-graph.

Algorithm <ref> terminates and Properties <ref> and <ref> hold throughout.

Observe that <ref> and <ref> hold when we
initialise the sets and put $m=0$ on line <ref>.  Assume
that we are in some step of the algorithm,
where <ref> and <ref> hold and we have
vertices $U=(u_1,\ldots,u_m)$ forming a $(2,3)$-path (this is true
because of <ref>).

Now we consider the next execution of the while loop.
We know from <ref> that $W_U$ contains exactly the vertices used
in the edges $u_iu_{i+1}(w)$ for $i=1,\dots,m-1$.  Either we extend
the path by an edge $u_mv$ or $u_mv(w)$ (lines <ref>
and <ref>) where $w$ is added to $W_U$ if needed
(line <ref>), or we remove an edge $u_{m-1}u_m$ or
$u_{m-1}u_m(w)$ (lines <ref> and <ref>) where $w$
is removed from $W_U$ if needed (line <ref>).
Therefore, <ref> still holds.

For <ref> it is easy to see that $T_i\subseteq V_i$ for $i\in[k]$, as in the beginning of the execution we have $T_i=V_i$ and no vertex is added to $T_i$.
Also, since every $w$ in $W_U$ comes from $T_1\cup\dots\cup T_k$, we have $W_U\subseteq T_1\cup\dots\cup T_{k-1}$ (see lines <ref> and <ref>).
Since every vertex of $S$ comes from $U$ (line <ref>) and every vertex of $U$ comes from $T_k\subset V_k$ (lines <ref>, <ref> and <ref>), which implies that $S$, $U\subset V_k$.
To prove that $|W_S| \le |S|$, it is enough to
observe that line <ref> can only be executed after an
execution of line <ref>.  Similarly, we have
$|W_U| \le m-1$ with $m \ge 2$, because line <ref> can only
be executed after an execution of line <ref>, and $|W_U|=0$
with $m=1$, because on line <ref> nothing is added to $W_U$.
Thus, <ref> also remains true.

It remains to show that the algorithm terminates.
In every execution of the while loop, either one vertex from $T_k\subseteq V_k$
is added to the path $U$ (lines <ref> and <ref>) or
moved from $U$ to $S$ (line <ref>).
Therefore, after at most $2|V_k|$ steps we have $T=\emptyset$, and the
algorithm terminates.

We are ready to prove the aforementioned result on $(2,3)$-graphs $F$ with no long $(2,3)$-paths.

Let $k$, $c$ and $n$ be positive integers and let $F=(V,E)$ be a $(2,3)$-graph on at least $5^{k-1}cn$ vertices.
If $F$ contains no $(2,3)$-path with $n$ vertices, then there exist disjoint sets
$V_1,\dots,V_{k} \subseteq V$ of size at least $cn$ such that no
edge from $E$ is a transversal and there is no edge $uv(w)$ with $u \in V_1 \cup \dots \cup V_{k-1}$ and $v,w \in V_{k}$.

We prove the result by induction on $k$.
For $k=1$ the result follows by putting $V_1=V(F)$.
Thus let $k\geq 1$ and assume the statement holds for $k$.

To prove the result for $k+1$, let $F$ be a $(2,3)$-graph on $5^k cn$ vertices which does not have a path of length $n$.
In particular, $F$ does not have a path of length $5n$, and by the assumption on $k$, there exist disjoint sets $V_1,\cdots,V_k$, each of size $5cn$, such that no edge is a transversal, and there is no edge $uv(w)$ with $u \in V_1 \cup \dots \cup V_{k-1}$ and $v,w \in V_{k}$.
We run Algorithm <ref> with input $F$, $k$ and  $V_1, \dots, V_{k}$.

First, we prove that at any point in the execution of the algorithm, no edge is a transversal with respect to $T_1\cup\dots\cup T_k, S$.
Suppose for a contradiction that at some point there is an edge which is a transversal.
Note that $T_k\subseteq V_k$ and $S\subseteq V_k$.
If $uv$ is this edge, then by the induction hypothesis and without loss of generality we have $u\in S$ and $v\in T_k$.
This implies that when $u$ was moved from $U$ to $S$ (line <ref>), the set $U$ could have been extended, which means that $T_\ext\neq \emptyset$ and line <ref> would not have been executed, a contradiction.
Now, assume $uv(w)$ is the transversal.
Since $T_i\subseteq V_i$ for $i\in [k]$ and $S\subseteq V_k$, we have $w\in T_1\cup\dots\cup T_{k-1}$.
Again, by the induction hypothesis and without loss of generality we have $u\in S$ and $v\in T_k$.
Similarly as when we have an edge $uv$, at the time $u$ was moved from $U$ to $S$, the set $U$ could have been extended, a contradiction.

Now we prove that at any point in the execution of the algorithm, there is no edge $uv(w)$ with $u \in T_1,\dots,T_{k-1},S$ and $v,w \in T_k$.
Suppose for a contradiction that at some point there is such edge $uv(w)$.
By the induction hypothesis we have $u\in S$ and $v,w\in T_k$, which again gives a contradiction as $U$ could have been extended.

Note that since $F$ has at least $5^{k-1}cn$ vertices and no $(2,3)$-path with $n$ vertices, we have $|S|=cn$ at some point of the execution of Algorithm <ref>.
Let $U$, $W_U$, $S$, $W_S$ and $T_1,\dots,T_k$ be the sets at that moment.
Note that $|W_S|\leq |S|= cn$ and, since there is no $n$-vertex $(2,3)$-path in $F$, we have $|U|$, $|W_U|\leq n$.
Therefore, $|T_i|\geq |V_i|-2cn\geq cn$ for $i\in [k-1]$ and $|T_k|\geq |V_k|-4cn \geq cn$.
Put $V_i' = T_i$ for $i\in [k-1]$, $V_k'=S$ and $V_{k+1}'=T_k$.
The sets $V_1',\dots,V_{k+1}'$ satisfies the requirements of the lemma.

§ PROOF OF THEOREM <REF>

In this section we prove our main theorem.  We first define the
following constants:

\begin{align*}
\ell:=17, \quad k:=2 \ell,\quad \eps:=1/(k+1),\quad t:=8k+40k^2+5,
\quad\text{and}\quad t':=r_2(K_t^{(3)}),
\end{align*}

where $r_s(K_t^{(3)}) := \min \{ n : K_n^{(3)} \rightarrow (K_t^{(3)})_s \}$
is the classical Ramsey-number for hypergraphs.
We start by obtaining a graph $G$ with bounded maximum degree and some
nice pseudorandom properties.  Let $a_{{\rm L}.\ref{lem:H2} }$ be
large enough to apply Lemma <ref> with $k$ and $\eps$ and set

\begin{align*}
c := \eps a_{{\rm L}.\ref{lem:H2} }\ell.
\end{align*}

Let $a_{{\rm L}.\ref{lem:transversal}} = 5^{k}$ and note that $a_{{\rm L}.\ref{lem:transversal}}$ is large enough to apply Lemma <ref> with $k+1$, $c$ and $n$ and set

\begin{align*}
a :=2 a_{{\rm L}.\ref{lem:transversal}}.
\end{align*}

Lemma <ref> applied with $\eps$ and $a$ provides a
constant $b$.  Let $n$ be sufficiently large.  Then, from
Lemma <ref> we know that there is a graph $G$ on $an$
vertices with maximum degree $b$ such that <ref> holds.
Fix such a graph $G$.

Now let $G^k(t')$ be the graph obtained from $G^k$ – the $k$-th
power of $G$ – by replacing every vertex by a $K_{t'}$ and every edge by a
$K_{t',t'}$.  Finally, $H$ is the $3$-uniform hypergraph with vertex
set $V(G^k(t'))$ and a triple of vertices $xyz$ is an edge in $H$ if
and only if $xyz$ forms a triangle in $G^k(t')$.  For every
$v \in V(G)$ we denote by $H(v)$ the corresponding cluster consisting
of a $K^{(3)}_{t'}$ in $H$.  We claim that
$H \rightarrow (P^{(3)}_{n})_2$.  Since $|V(H)| = a t' n$ and
$\Delta(H) \le b^{2k+2} t'^3$, this would prove Theorem <ref>.

The rest of the proof is devoted to proving
that $H \rightarrow (P^{(3)}_{n})_2$.  Fix a $2$-colouring of the
triples of $H$. As $t' \ge r_2(K^{(3)}_t)$ for every $v \in V(G)$
the cluster $H(v)$ either contains a red or blue copy of $K^{(3)}_t$.
W.l.o.g. there is a set of vertices $V \subseteq V(G)$ with
$|V| \ge an/2 = a_{{\rm L}.\ref{lem:transversal}} n$ such that for
all $v \in V$ the cluster $H(v)$ contains a blue copy of $K^{(3)}_t$,
which we denote by $H'(v)$.
We let $H' \subseteq H$ be the $3$-graph induced by the clusters
$H'(v)$ for $v \in V$.

We will define an auxiliary $(2,3)$-graph $F$ on the vertex
set $V$, whose edges will indicate that we can walk between the clusters
using blue triples of $H'$.  Formally, for $u,v \in V$ a $(2,2)$-connector
between the clusters $H'(u)$ and $H'(v)$ consists of four vertices
$x_1,x_2 \in H'(u)$ and $y_1,y_2 \in H'(v)$ such that $x_1x_2y_1$ and
$y_1y_2x_1$ are triples of $H$.  Similarly, for $u,v,w \in V$ a
$(2,1,2)$-connector between the clusters $H'(u)$ and $H'(v)$
through $H'(w)$ consists of five vertices $x_1,x_2 \in H'(u)$,
$z \in H'(w)$, and $y_1,y_2 \in H'(v)$ such that $x_1x_2z$, $x_1zy_1$,
and $zy_1y_2$ are triples of $H$; see Figure <ref>.  We
then define a $(6,6)$-connector ($(6,3,6)$-connector) between
$H'(u)$ and $H'(v)$ (through $H'(w)$) as the
disjoint union of three $(2,2)$-connectors
($(2,1,2)$-connectors) between $H'(u)$ and $H'(v)$ (through $H'(w)$).

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[draw,circle,inner sep=3pt,fill] at (1,.5) ;
[draw,circle,inner sep=3pt,fill] at (0,1.5) ;
[draw,circle,inner sep=3pt,fill] at (3,.5) ;
[draw,circle,inner sep=3pt,fill] at (4,1.5) ;
[draw,circle,inner sep=3pt,fill] at (2,2.5) ;

at (.7,0) $x_1$;
at (-.5,1.2) $x_2$;
at (3.3,0) $y_1$;
at (4.5,1.2) $y_2$;
at (2,3) $z$;

at (-.4,.5) $H'(u)$;
at (4.4,.5) $H'(v)$;
at (2,3.5) $H'(w)$;

A $(2,2)$-connector and a $(2,1,2)$-connector.

Let $F$ be a $(2,3)$-graph on the vertex set $V$ with the following
two types of edges:

* $uv\in E(F)$ if and only if there is a $(6,6)$-connector in blue
between the corresponding clusters $H'(u)$ and $H'(v)$;

* $uv(w)\in E(F)$ if and only if there is a $(6,3,6)$-connector in
blue between the corresponding clusters $H'(u)$ and $H'(v)$ through

Suppose that $F$ contains a $(2,3)$-path on $n$ vertices
$v_1,\dots,v_n$ with the $w_i$ vertices all distinct for the $3$-edges.
We can turn this $(2,3)$-path into a blue tight path $P^{(3)}_n$ in $H$ as follows.
First, by following the $(2,3)$-path for $i = 1,\dots,n-1$, we choose a $(2,2)$-connector between $H'(v_i)$ and $H'(v_{i+1})$ if $v_iv_{i+1} \in E(F)$ or a $(2,1,2)$-connector between $H'(v_i)$ and $H'(v_{i+1})$ through $H'(w_i)$ if $v_iv_{i+1}(w_i) \in E(F)$ in such a way that they are all pairwise vertex-disjoint.
This is possible, because we have $(6,6)$-connectors and $(6,3,6)$-connectors available and two vertices $y_1$ and $y_2$ from the previous connector can occupy at most two out of the three disjoint copies of connectors that are provided.
Then, for each of the clusters with $y_1$, $y_2$ and $x_1$, $x_2$ the vertices of the connectors within that cluster, we use the edges $y_1y_2x_2$ and $y_2x_2x_1$ to connect both connectors.
As the connectors only use blue edges and all edges within the clusters are blue this is a tight path only using blue edges.
Thus, in this case, we are able to obtain a blue $P^{(3)}_n$ in $H$ and we are done.

We assume that $F$ contains no $(2,3)$-path on $n$ vertices.
From Lemma <ref>, there exist pairwise disjoint sets
$V_1,\dots,V_{k+1} \subseteq V$ of size at least $cn$ such that no
edge from $E(F)$ is a transversal.

We may assume that all these sets $V_j$ have size exactly $cn$.
Let $G'=G[V_1\cup\dots\cup V_{k+1}]$ and set $m := \ell n$.

We now want to find a path $P_m$ alternating through $V_1,\dots,V_{k+1}$
with edges in $G'\subseteq G$ using Lemma <ref>. Since
$c=\eps a_{{\rm L}.\ref{lem:H2} }\ell$ and $\eps=1/(k+1)$, we have
$|V(G')| = (k+1) c n = a_{{\rm L}.\ref{lem:H2}}\ell n =a_{{\rm
L}.\ref{lem:H2}}m$.  Also, we have
$|V_i| =  cn = \eps a_{{\rm L}.\ref{lem:H2}} m$ for $1\leq i\leq k+1$.
As $G'$ is an induced
subgraph of $G$ and property <ref> holds in $G$, property [prop:exist](P1$_m$) does hold for $G'$.
Therefore, by Lemma <ref>, we conclude that there is a path
$P_m=P_{\ell n}$ with vertices alternating through $V_1,\dots,V_{k+1}$
and with edges in $G'\subseteq G$.

This path $P_{\ell n}$ gives us the $k$th power $P^k_{\ell n}$
in $G^k$.  By the choice of $V_1,\dots,V_{k+1}$ no edge of
$P^k_{\ell n}$ is from $E(F)$ and also no triangle in $P^k_{\ell n}$
induces an edge $uv(w) \in E(F)$.  It remains to turn this
$P^k_{\ell n}$ into a red $P^{(3)}_n$ in $H'$.

If there is a $P_{\ell n}^k$ in $G^k$ that does not contain any
edges from $F$, then there is a red $P^{(3)}_n$ in $H'$.

Let $P_{\ell n}^k=(v_1,\dots,v_{\ell n})$ and recall $H'(v_i)$ is the
cluster in $H'$ corresponding to the vertex $v_i$ for
$i=1,\dots,\ell n$.  We want to remove all vertices of $H'$ which
belong to blue $(2,2)$-connectors and $(2,1,2)$-connectors from
clusters along edges and triangles of $P_{\ell n}^k$.
In $P_{\ell n}^k$ every vertex $v_i$ is incident to at most $2k$ other vertices in
$\{v_1,\dots,v_{\ell n}\}$ ($2k$ is the maximum degree of the $v_i$
in $P^k_{\ell n}$).  Also, every $v_i$ is contained in at most
$4k^2$ triangles of $P_{\ell n}^k$ together with two other vertices in
$\{v_1,\dots,v_{\ell n}\}$.

Let $u$ and $v$ be neighbours in $P_{\ell n}^k$.  Since there is no
blue $(6,6)$-connector between $H'(u)$ and $H'(v)$, there are at most
two $(2,2)$-connectors that do not overlap between $H'(u)$ and
$H'(v)$, which can both be deleted by removing at most $4$ vertices in each
cluster.  Let $u$, $v$, and $w$ be vertices that form a triangle in
$P_{\ell n}^k$.  Since there is no blue $(6,3,6)$-connector between
$H'(u)$, $H'(v)$, and $H'(w)$, there are at most six
$(2,1,2)$-connectors that do not overlap, two for each possibility to
place the single vertex.  These can be deleted by removing at most
$10$ vertices from each cluster.

By the above argument, we have to delete at most
$4(2k) + 10(4 k^2) \le t - 5$ vertices from every cluster to get rid
of all $(2,2)$-connectors and $(2,1,2)$-connectors.  Let
$H^*(v_i) \subseteq H'(v_i)$ be the remainder of the corresponding
cluster in $H'$, and note that $|H^*(v_i)| \ge 5$ for
$i=1,\dots,\ell n$.

A tuple $(u,v)$ is an end-tuple of a tight path with at
least $4$ vertices if $u$ and $v$ are consecutive vertices in the path
and $u$ is contained in exactly two edges and $v$ is contained in
exactly one.
The two tuples $(u,v)$ and $(v,w)$ are the end-tuples of the tight path $(u,v,w)$ of
length $3$.
Furthermore, every tuple $(u,v)$ is an end-tuple of the tight
path $(u,v)$ of length $2$.

For $i=1,\dots,n-1$ we say that the quadruple $(u_1,u_2,w_1,w_2)$ satisfies
property $Q_i$ if the following conditions hold:

* $u_1,u_2,w_1,w_2$ are distinct vertices from $H'$ such that the pairs
$u_1,u_2$ and $w_1,w_2$ are in clusters
$H_r,H_s \in \{H^*(v_{(i-1)\ell +1}), \dots, H^*(v_{i\ell})\}$, respectively, where
$r\neq s$;

* each of $(u_1,u_2)$ and $(w_1,w_2)$ is an end-tuple of a
red tight path of length at least $i+1$ with vertices in
$H^*(v_1)\cup \dots\cup H^*(v_{i\ell})$.

To prove Claim <ref>, i.e., construct $P^{(3)}_n$ in red,
it is then sufficient to construct a quadruple satisfying property
$Q_{n-1}$.  We will construct this quadruple inductively.  The base case
$Q_1$ asks for two paths of length $2$ and, therefore, it is enough to
choose any pair $u_1,u_2$ from $H^*(v_{1})$ and $w_1,w_2$ from
$H^*(v_{2})$.  Therefore, the following is immediate:

\begin{equation}
\label{lem:basecase}
\text{There exists a quadruple $(u_1,u_2,w_1,w_2)$ for which $Q_1$
\end{equation}

We will inductively find a quadruple $(u_1,u_2,w_1,w_2)$ satisfying
property $Q_i$ for every $i=2,\dots,n-1$.  Ultimately, after $n$ steps,
this gives us $P^{(3)}_n$ in red in $H'$.  Suppose $(u_1,u_2,w_1,w_2)$
satisfies property $Q_{i-1}$ for some $1<i\leq n-1$.  In the inductive
step, we obtain $(x_1,x_2,y_1,y_2)$ satisfying property $Q_i$ by
extending one of the paths ending in $(u_1,u_2)$ or $(w_1,w_2)$ to get
two longer paths ending in $(x_1,x_2)$ and $(y_1,y_2)$.  This mainly
relies on the absence of $(2,2)$-connectors and $(2,1,2)$-connectors
in blue and that there are $\ell$ clusters $H^*(v_{(i-1)\ell+1}),\dots,H^*(v_{i \ell})$
to choose $x_1,x_2,y_1,y_2$ from. As $k \ge 2\ell$, all edges are present between
these clusters and the clusters $H_r$ and $H_s$ containing $u_1,u_2$
and $w_1,w_2$, respectively.

Let $1< i \le n-1$ and suppose $(u_1,u_2,w_1,w_2)$ satisfies
property $Q_{i-1}$.  Then there is a quadruple $(x_1,x_2,y_1,y_2)$
that satisfies property $Q_{i}$.

We will find $(x_1,x_2,y_1,y_2)$ following the strategy sketched
above.  Since there are no blue $(2,2)$-connectors between the
clusters corresponding to $u_1,u_2$ and $w_1,w_2$, and  all
possible triples between these clusters are edges in $H'$, then either the triple
$u_1u_2w_2$ or the triple $w_1w_2u_2$ is red, say, w.l.o.g.,
$u_1 u_2 w_2$ is red. We let the red path of length at least $i$ that ends in $(u_1,u_2)$
be called $P_\red$ and note that the triple $u_1 u_2 w_2$ already
extends this path. We
will show that it is possible to further extend this path to obtain two
longer red tight paths with ends $(x_1,x_2)$ and
$(y_1,y_2)$, respectively, such that $(x_1,x_2,y_1,y_2)$ satisfies property

Notice that, as $\ell \ge 17$, by the pigeonhole principle there are
nine sets $X_1,\dots,X_9$ each contained in a different cluster from
$H^*(v_{(i-1)\ell +1}), \dots, H^*(v_{i\ell})$ and of size
$|X_j| \ge 3$ for $j \in [9]$ (here we use that $|H^*(v_i)|\ge5$)
such that either

\begin{align}
\label{eq:edgesXpresent}
\text{for all $j \in [9]$ and every $x \in X_j$ the triple $u_2w_2x$ is red }
\end{align}

\begin{align}
\label{eq:edgesXnotpresent}
\text{for all $j \in [9]$ and every $x \in X_j$ the triple $u_2w_2x$ is blue.}
\end{align}

We first consider the case where (<ref>) holds.  It
is enough to assume that we have $X_1,\dots,X_5$ and $|X_j|\ge 2$
for $j \in [5]$.  If there are two sets $X,Y$ from $X_1,\dots,X_5$
and $x_1,x_2 \in X$, $y_1,y_2 \in Y$ such that the triples $w_2x_1x_2$
and $w_2y_1y_2$ are red, then the quadruple $(x_1,x_2,y_1,y_2)$
satisfies property $Q_i$ as we can obtain two longer red paths by
extending $P_\red$ following $u_1u_2w_2x_1x_2$ and
$u_1u_2w_2y_1y_2$, respectively.  Otherwise, there are two sets
$X,Y$ and $x_1,x_2 \in X$, $y_1,y_2 \in Y$ such that the triples
$w_2x_1x_2$ and $w_2y_1y_2$ are blue.  As there is no blue
$(2,1,2)$-connector, the triple $w_2x_1y_1$ is red.  There is no blue
$(2,2)$-connector between the corresponding clusters, so either the
triple $x_1y_1y_2$ or $y_1x_1x_2$ is red, say w.l.o.g. $x_1y_1y_2$ is
red.  This extends $P_\red$ by following $u_1u_2w_2x_1y_1y_2$ and
gives the end-tuple $(y_1,y_2)$.  Repeating the same argument, which
is possible, because there were five sets available (sets
$X_1,\dots,X_5$), we get an end-tuple $(z_1,z_2)$ that extends
$P_\red$ to a longer red path and, thus, a quadruple
$(y_1,y_2,z_1,z_2)$ satisfying $Q_i$.

In the case where (<ref>) holds we proceed as
follows.  As for all $j \in [9]$ there is no blue
$(2,1,2)$-connector between the clusters of $u_1,u_2$ and $w_1,w_2$
and $X_j$, we have for every $x \in X_j$ that either the triple
$w_1w_2x$ or the triple $u_1u_2x$ is red.  Then we can assume by the
pigeonhole principle w.l.o.g. (we will not use the triple $u_1u_2w_1$)
that there are sets $X_j' \subseteq X_j$ with $|X_j'|\ge 2$ for
$j \in [5]$ (here we use that $|X_j|\ge 3$) such that

\begin{align*}
\text{for all $j \in [5]$ and every $x \in X_j'$ the triple $w_1w_2x$ is red.}
\end{align*}

Now we can continue exactly as in the case
where (<ref>) holds, with $u_1u_2$ replaced by
$w_1w_2$ throughout and extending the path with end-tuple
$(w_1,w_2)$.  Observing that the red tight paths that we have
constructed have length at least $i+1$, we see that
Fact <ref> is proved.

Fact <ref> together with (<ref>)
finishes the proof of Claim <ref> and hence the
proof of Theorem <ref> is complete.

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Jie Han

School of Mathematics and Statistics

Beijing Institute of Technology

Beijing, China

Yoshiharu Kohayakawa

Instituto de Matemática e Estatística

Universidade de São Paulo

Rua do Matão 1010, 05508-090 São Paulo, Brazil

Shoham Letzter

Department of Mathematics

University College London

Gower Street, London WC1E 6BT

Guilherme Oliveira Mota

Instituto de Matemática e Estatística

Universidade de São Paulo

Rua do Matão 1010, 05508-090 São Paulo, Brazil

Olaf Parczyk

London School of Economics

Department of Mathematics

Houghton Street, London, WC2A 2AE, UK