# Quantum Hall effect in topological Dirac semimetals modulated by the
Lifshitz transition of the Fermi arc surface states

Tao-Rui Qin1 Zhuo-Hua Chen1 Tian-Xing Liu1 Fu-Yang Chen1 Hou-Jian Duan1,2
Ming-Xun Deng1,2<EMAIL_ADDRESS>Rui-Qiang Wang1,2
<EMAIL_ADDRESS>1Guangdong Provincial Key Laboratory of Quantum
Engineering and Quantum Materials, School of Physics, South China Normal
University, Guangzhou 510006, China  2Guangdong-Hong Kong Joint Laboratory of
Quantum Matter, Frontier Research Institute for Physics, South China Normal
University, Guangzhou 510006, China

###### Abstract

We investigate the magnetotransport of topological Dirac semimetals (DSMs) by
taking into account the Lifshitz transition of the Fermi arc surface states.
We demonstrate that a bulk momentum-dependent gap term, which is usually
neglected in study of the bulk energy-band topology, can cause the Lifshitz
transition by developing an additional Dirac cone for the surface to prevent
the Fermi arcs from connecting the bulk Dirac points. As a result, the Weyl
orbits can be turned off by the surface Dirac cone without destroying the bulk
Dirac points. In response to the surface Lifshitz transition, the Weyl-orbit
mechanism for the 3D quantum Hall effect (QHE) in topological DSMs will break
down. The resulting quantized Hall plateaus can be thickness-dependent,
similar to the Weyl-orbit mechanism, but their widths and quantized values
become irregular. Accordingly, we propose that apart from the bulk Weyl nodes
and Fermi arcs, the surface Lifshitz transition is also crucial for realizing
stable Weyl orbits and 3D QHE in real materials.

##  I introduction

Topological semimetals are novel quantum states of matter, in which the
conduction and valence bands cross near the Fermi level at certain discrete
momentum points or linesArmitage  _et al._ (2018); Neupane  _et al._ (2014);
Wang  _et al._ (2013, 2012); Liu  _et al._ (2014); Yang and Nagaosa (2014);
Wan  _et al._ (2011); Zhang  _et al._ (2016, 2017); Wang  _et al._ (2017). The
gap-closing points or lines are protected either by crystalline symmetry or
topological invariantsGorbar  _et al._ (2015); Kargarian  _et al._ (2016). A
topological Dirac semimetal (DSM) hosts paired gap-closing points, referred to
as the Dirac points, which are stabilized by the time-reversal, spatial-
inversion and crystalline symmetries. By breaking the time-reversal or
spatial-inversion symmetry, a single Dirac point can split into a pair of Weyl
nodes of opposite chiralitiesNielsen and Ninomiya (1983); Volovik (2003),
leading to the topological transition from a Dirac to a Weyl semimetalRaza
_et al._ (2019); Zyuzin  _et al._ (2012); Goswami and Tewari (2013); Han  _et
al._ (2018); Deng  _et al._ (2017); Chen  _et al._ (2019). Accompanied with
the bulk topological transition, topological states which are protected by the
quantized Chern flux will emerge in the surface to connect the split Weyl
nodes, known as the Fermi-arc surface statesWan  _et al._ (2011).

In topological DSMs, such as A3Bi (A = Na,K,Rb)Wang  _et al._ (2012); Liu  _et
al._ (2014) and Cd2As3Wang  _et al._ (2013); Neupane  _et al._ (2014), the
Weyl nodes at the same Dirac point, belonging to different irreducible
representations, cannot be coupled and have to seek for a partner from the
other Dirac point. As a consequence, the two Dirac points including two pairs
of Weyl nodes are connected by two spin-polarized Fermi arcsWang  _et al._
(2012, 2013); Liu  _et al._ (2014); Yang and Nagaosa (2014); Wan  _et al._
(2011). The Fermi-arc surface states are the most distinctive observable
spectroscopic feature of topological semimetals. However, their observation is
sometime limited by spectroscopic resolutions. There have been searching for
alternative smoking-gun features of topological semimetals, such as by means
of transport phenomenaAli  _et al._ (2014); Shekhar  _et al._ (2015);
Parameswaran  _et al._ (2014). Many interesting transport properties have been
revealed in topological semimetals, for example, chiral anomaly induced
negative magnetoresistanceXiong  _et al._ (2015); Li  _et al._ (2015); Deng
_et al._ (2019a, b), Weyl-orbit related quantum oscillationsPotter  _et al._
(2014); Moll  _et al._ (2016), Berry’s phase $\pi$ related Aharonov-Bohm
effectLin  _et al._ (2017); Wang  _et al._ (2016), bulk-surface interference
induced Fano effectWang  _et al._ (2018), topological pumping effectDeng  _et
al._ (2022), etc.

Recently, 3D quantum Hall effect (QHE) because of the Fermi arcs was proposed
in Weyl semimetalsWang  _et al._ (2017) and has led to an explosion of
theoreticalLi  _et al._ (2020); Ma  _et al._ (2021); Geng  _et al._ (2021);
Chang  _et al._ (2021); Chang and Sheng (2021); Lu (2018); Li  _et al._
(2021); Qin  _et al._ (2020); Gooth  _et al._ (2023); Wang and Cai (2023); Du
_et al._ (2021) and experimentalUchida  _et al._ (2017); Schumann  _et al._
(2018); Zhang  _et al._ (2019); Tang  _et al._ (2019); Lin  _et al._ (2019)
activities in the field of condensed matter physics. In a Weyl semimetal slab,
the Weyl orbit which consists of Fermi arcs from opposite surfaces can support
the electron cyclotron orbit for the QHE, making the 3D QHE available. The 3D
QHE has been observed experimentally in topological DSMsUchida  _et al._
(2017); Schumann  _et al._ (2018); Zhang  _et al._ (2019); Tang  _et al._
(2019); Lin  _et al._ (2019) and the one from the Weyl-orbit mechanism is
demonstrated to be thickness-dependent Zhang  _et al._ (2019). However, for
the topological DSMs, a single surface with two Fermi arcs can also support a
complete Fermi loop required by the QHE, which can compete with the Weyl-orbit
mechanism. The same-surface Fermi loop is not stable and can be deformed by
bulk perturbationsKargarian  _et al._ (2016). In real materials, the bulk
perturbations are inevitable.

As we will show, when a bulk momentum-dependent gap term is included, Lifshitz
transition can happen for the Fermi arc surface states, in which the double
Fermi arcs on the DSM surface can be continuously deformed into a closed Fermi
loop and separate from the bulk Dirac points. The Lifshitz transition involves
a change of the Fermi surface topology that is not connected with a change in
the symmetry of the latticeLifshitz (1960); Okada  _et al._ (2013); Zeljkovic
_et al._ (2014); Wu  _et al._ (2015); Ding  _et al._ (2021). Therefore, the
Lifshitz transition can take place without destroying the topology of the bulk
energy band. A natural question in this regard is how the deformation of the
Fermi arcs influences the 3D QHE of topological DSMs, especially when the
Fermi arcs, as key ingredients for the Weyl orbits, breaks free from the bulk
Dirac points.

In this paper, we investigate the QHE in topological DSMs by taking into
account the surface Lifshitz transition, which can be modulated by a bulk
momentum-dependent gap term. It is demonstrated that while the bulk Dirac
points are robust against the momentum-dependent gap term, the surface can
develop an additional 2D Dirac cone, which deforms the surface Fermi arcs from
a curve to some discrete points and further to a Fermi loop coexisting with
the bulk Dirac points. During this process, the bulk topological properties do
not change, but the Weyl orbits can be turned off. The joint effect of the
Weyl orbits and surface Lifshitz transition can make the QHE quite
complicated. We find that when the Weyl orbits are broken by the surface Dirac
cone, the bulk and surface states can form the Landau levels (LLs) and
contribute the QHE, independently. The resulting Hall plateaus are sensitive
to the thickness of the sample, but their widths and quantized values are
irregularly distributed. The rest of this paper is organized as follows. In
Sec. II, we introduce the model Hamiltonian and bulk spectrum. The Lifshitz
transition of the Fermi arcs and the LLs are analyzed in Sec. III and Sec. IV,
respectively. The QHE is studied in Sec. V and the last section contains a
short summary.

##  II Hamiltonian and bulk spectrum

We begin with a low-energy effective Hamiltonian for the topological DSMs

$\mathcal{H}(\bm{k})=\varepsilon_{\bm{k}}+\lambda(k_{x}\sigma_{z}\tau_{x}-k_{y}\tau_{y})+m_{\bm{k}}\tau_{z}+\Lambda(\bm{k})$
(1)

with $m_{\bm{k}}=m_{0}-m_{1}k_{z}^{2}-m_{2}k_{\parallel}^{2}$ and
$\varepsilon_{\bm{k}}=c_{0}+c_{1}k_{z}^{2}+c_{2}k_{\parallel}^{2}$, where
$k_{\parallel}=\sqrt{k_{x}^{2}+k_{y}^{2}}$ and $\sigma_{x,y,z}$
($\tau_{x,y,z}$) is the Pauli matrix acting on the spin (orbital parity)
degree of freedom. This model has been widely-adopted to capture the
topological properties of topological DSMs Cd3As2Wang  _et al._ (2013) and
A3Bi (A = Na, K, Rb)Wang  _et al._ (2012). In the absence of
$\Lambda(\bm{k})$, $\left[\sigma_{z},\mathcal{H}(\bm{k})\right]=0$ and the
topological DSMs characterized by Hamiltonian (1) can be viewed as two
superposed copies of a Weyl semimetal with two Weyl nodes, which possesses two
sets of surface Fermi arcs in the surface Brillouin zone, as illustrated by
Figs. 1(a)-(b) and (e)-(f). $\Lambda(\bm{k})$ mixes the eigenstates of
opposite spins away from the Dirac points and plays the role of a momentum-
dependent gap term, whose form is determined by the crystal symmetries.

Specifically, for a DSM with fourfold rotational symmetry, such as Cd3As2, the
momentum-dependent gap term can take the form $\Lambda(\bm{k})=\alpha
k_{z}\left(k_{+}^{2}\sigma_{-}+k_{-}^{2}\sigma_{+}\right)\tau_{x}/2$ with
$k_{\pm}=k_{x}\pm ik_{y}$ and $\sigma_{\pm}=\sigma_{x}\pm i\sigma_{y}$.
Diagonalizing Hamiltonian (1) yields the continuum bulk spectrum

$E_{\pm}(\bm{k})=\varepsilon_{\bm{k}}\pm\sqrt{\lambda^{2}k_{\parallel}^{2}+m_{\bm{k}}^{2}+\alpha^{2}k_{\parallel}^{4}k_{z}^{2}},$
(2)

from which we can determine the energy location $E_{D}=c_{0}+c_{1}k_{w}^{2}$
and momentum locations $\bm{K}_{\pm}=(0,0,\pm k_{w})$ of the bulk Dirac points
with $k_{w}=\sqrt{m_{0}/m_{1}}$. As $\varepsilon_{\bm{k}}$ possesses the
symmetries of $m_{\bm{k}}$, it does not qualitatively change the Dirac
spectrum in the bulk, but introduces an asymmetry between the positive
(electrons) and negative (holes) energy branches and, consequently, breaks the
particle-hole symmetry. The electron-hole asymmetry will curve the Fermi arcs,
which was demonstrated to be crucial for the LLs around the Weyl nodesWang
_et al._ (2017). While $\Lambda(\bm{k})$ can profoundly change the spectrum of
quasiparticles for sufficiently large $k_{\parallel}$, it preserves all the
symmetries of the crystal structure and vanishes at the Dirac points.
Therefore, the bulk Dirac points are robust against $\Lambda(\bm{k})$, as seen
from Eq. (2) that both the momentum and energy locations of the Dirac points
are regardless of $\alpha$. For this reason, $\Lambda(\bm{k})$ was usually
treated as a bulk perturbation, which does not destroy the bulk topology.

Figure 1:  (a)-(d) The dispersion (red) by diagonalizing Hamiltonian (12) for
(a) $\alpha=0$, $c_{1}=c_{2}=0$, (b) $\alpha=0$, $c_{1}=3c_{2}=0.15$, (c)
$\alpha=0.5$, $c_{1}=c_{2}=0$, and (d) $\alpha=0.5$, $c_{1}=3c_{2}=0.15$, with
$k_{x}=0$ and $E_{D}=0$ denoted by the blue-dashed lines. The green dotted
curves are determined from Eq. (9), self-consistently. (e)-(h) The
$\bm{k}$-resolved DOSs corresponding to (a)-(d) at $E_{F}=E_{D}$. The rest
parameters are set as $\lambda=0.5$, $k_{w}=\pi/4$, $m_{0}=\pi^{2}/32$,
$m_{2}=0.5$ and $N_{y}=50$.

For the states close to the bulk Dirac points, the quasiparticles can be
described by linearizing Hamiltonian (1) around $\bm{K}_{\pm}$, such that
$\Lambda(\bm{k})$ can be neglected. However, for a topological DSM slab, the
Fermi arcs, which connect the bulk Dirac points separated far away in momentum
space, can extend to large $\bm{k}$ where the spectrum can be dramatically
modified by the momentum-dependent gap term. In the following, we elucidate
that, in response to the momentum-dependent gap term, the surface states of
topological DSMs will experience a Lifshitz transition, during which process,
the Fermi arcs can exist without connecting the bulk Dirac points.

Figure 2: The LLs for (a) $\alpha=0$, $c_{1}=c_{2}=0$, (b) $\alpha=0$,
$c_{1}=3c_{2}=0.15$, (c) $\alpha=0.5$, $c_{1}=c_{2}=0$, and (d) $\alpha=0.5$,
$c_{1}=3c_{2}=0.15$. (e)-(h) The probability density $|\psi_{k_{z},\mu}|^{2}$
for the blue LLs in (a)-(d), in which $\psi_{k_{z},\mu}$ is the $\mu$-th LL’s
wavefunction. Here, we set $2\pi\ell_{B}^{2}=N_{x}$, $k_{z}=\pi$,
$k_{w}=\pi/2$, $N_{x}=40$ and other parameters the same as Fig. 1.

##  III Lifshitz transition of the Fermi arcs

To better understand how the Fermi arc surface states evolve with the
momentum-dependent gap term, we consider a topological DSM slab with open
boundaries at $y=\pm L/2$ and derive the surface states for the $x$-$z$ plane.
For such a finite-size system, $k_{y}$ in Eq. (1) is not longer a good quantum
number and should be replaced with the operator $-i\partial_{y}$. In the
spirit of the perturbation theory, we construct an unperturbed surface basis
by the surface wavefunctions without $\Lambda(\bm{k})$ and the effective
Hamiltonian for the surface states can be obtained by projecting Eq. (1) onto
the unperturbed surface basis. For the convenience of discussion, we assume
$m_{2}>|c_{2}|$ and select $E_{D}$ as potential energy zero, i.e.,
$c_{0}=-c_{1}k_{w}$.

By setting $\alpha=0$ and solving the differential equation
$\mathcal{H}$$(k_{x},-i\partial_{y},k_{z})\Psi_{k_{x},k_{z}}(y)=E\Psi_{k_{x},k_{z}}(y)$
with the open boundary conditions $\Psi_{k_{x},k_{z}}(\pm L/2)=0$, we can
evaluate the unperturbed surface wavefunctions as
$\Psi_{k_{x},k_{z}}^{\beta,s}(y)=\left(\frac{1+s}{2},\frac{1-s}{2}\right)^{T}\otimes\Psi_{\beta}(y)$,
which are spin-resolved, with $s=\pm 1$ being the eigenvalue of $\sigma_{z}$
and

$\Psi_{\beta}(y)=\left(\begin{array}[c]{c}\beta\cos\frac{\vartheta}{2}\\\
-\sin\frac{\vartheta}{2}\end{array}\right)\frac{e^{\kappa_{-}(\beta
y-\frac{L}{2})}-e^{\kappa_{+}(\beta y-\frac{L}{2})}}{\sqrt{N_{\beta}}}.$ (3)

Here, $\beta=\pm$__ corresponds to the surface at $y=\pm L/2$,
$N_{\beta}=\int_{-\frac{L}{2}}^{\frac{L}{2}}|e^{\kappa_{-}(\beta
y-\frac{L}{2})}-e^{\kappa_{+}(\beta y-\frac{L}{2})}|^{2}dy$ is the
normalization coefficient and

$\vartheta=\tan^{-1}\left[\frac{2\lambda\left(m_{2}-c_{2}\right)\left(\kappa_{+}+\kappa_{-}\right)}{\lambda^{2}-\left(m_{2}-c_{2}\right)^{2}\left(\kappa_{+}+\kappa_{-}\right)^{2}}\right],$
(4)

in which $\kappa_{\pm}$ is a solution to $E_{\pm}(k_{x},-i\kappa,k_{z})=E$,
reading as

$\kappa_{\pm}=\sqrt{k_{x}^{2}+\frac{\zeta_{\lambda}\pm\left(\zeta_{\lambda}^{2}-\zeta_{+}\zeta_{-}\right)^{1/2}}{m_{2}^{2}-c_{2}^{2}}}$
(5)

with $\zeta_{\lambda}=\frac{\lambda^{2}-\zeta_{+}+\zeta_{-}}{2}$ and

$\zeta_{\pm}=\left(m_{2}\pm c_{2}\right)\left[\left(m_{1}\mp
c_{1}\right)\left(k_{w}^{2}-k_{z}^{2}\right)\mp E\right].$ (6)

The surface states are confined within the region defined by
$\operatorname{Re}(\kappa_{\pm})>0$. Subsequently, by performing the
projection operation

$\mathcal{H}_{\mathrm{surf}}^{\beta,ss^{\prime}}=\langle\Psi_{k_{x},k_{z}}^{\beta,s}(y)|\mathcal{H}(k_{x},-i\partial_{y},k_{z})|\Psi_{k_{x},k_{z}}^{\beta,s^{\prime}}(y)\rangle,$
(7)

we can obtain the effective surface Hamiltonian

$\mathcal{H}_{\mathrm{surf}}^{\beta}\left(k_{x},k_{z}\right)=\tilde{\varepsilon}_{\bm{k}}-\beta\sin\vartheta\left(\lambda
k_{x}\sigma_{z}+\tilde{\alpha}k_{z}\sigma_{x}\right),$ (8)

where $\tilde{\alpha}=\alpha\left(k_{x}^{2}-\kappa_{+}\kappa_{-}\right)$,
$\tilde{\varepsilon}_{\bm{k}}=\tilde{c}_{1}\left(k_{z}^{2}-k_{w}^{2}\right)+\tilde{c}_{2}\left(k_{x}^{2}+\kappa_{+}\kappa_{-}\right)$
and $\tilde{c}_{l}=c_{l}-m_{l}\cos\vartheta$ with $l=1,2$.

As shown by Eq. (8), the surface Hamiltonian exhibits a 2D Dirac structure
with spin-momentum locking, which resembles the surface states for 3D
topological insulators. However, as different from the 3D topological
insulators, the bulk spectrum here is also gapless, namely, the surface Dirac
point can coexist with the bulk Dirac points. By diagonalizing Eq. (8), we can
arrive at

$E_{\eta}^{\beta}\left(k_{x},k_{z}\right)=\tilde{\varepsilon}_{\bm{k}}-\eta\beta\sin\vartheta\sqrt{\lambda^{2}k_{x}^{2}+\tilde{\alpha}^{2}k_{z}^{2}},$
(9)

where $\eta=\pm 1$ labels the conduction/valence band. It should be noted
that, because $\kappa_{\pm}$ is energy dependent, Eq. (9) is only a formal
solution for the surface spectrum. The exact surface dispersion involves a
self-consistent calculation of Eq. (9) via replacing $E\rightarrow
E_{\eta}^{\beta}\left(k_{x},k_{z}\right)$ in Eq. (6). The numerical results of
the surface dispersion are presented by the green dotted curves in Figs.
1(a)-(d).

In order to show the bulk states and surface Fermi arcs, simultaneously, we
evaluate the $\bm{k}$-resolved density of states (DOSs) for a given energy $E$
through

$\rho\left(E,k_{x},k_{z}\right)=-\frac{1}{\pi}\operatorname{Im}\mathrm{Tr}\left(\frac{1}{E+i0^{+}-H}\right),$
(10)

in which
$H=\sum_{\bm{k}}c_{\bm{k}}^{{\dagger}}\mathcal{H}_{\bm{k}}^{\mathrm{tb}}c_{\bm{k}}$
is the tight-binding Hamiltonian corresponding to Eq. (1). Here,
$c_{\bm{k}}^{{\dagger}}$ ($c_{\bm{k}}$) is the fermion creation (annihilation)
operator and

$\displaystyle\mathcal{H}_{\bm{k}}^{\mathrm{tb}}$
$\displaystyle=M_{\bm{k}}+\lambda\left(\sin k_{x}\sigma_{z}\tau_{x}-\sin
k_{y}\tau_{y}\right)-2\alpha\sin k_{z}$ $\displaystyle\times\left[\left(\cos
k_{x}-\cos k_{y}\right)\sigma_{x}-\sin k_{x}\sin
k_{y}\sigma_{y}\right]\tau_{x}$ (11)

is the single-particle Hamiltonian obtained from Eq. (1) via the transforms
$k_{a}\rightarrow\sin k_{a}$ and $k_{a}^{2}\rightarrow 2-2\cos k_{a}$, with
$a=x,y,z$, $M_{\bm{k}}=f_{1}\left(\cos k_{z}-\cos k_{w}\right)+f_{2}\left(\cos
k_{x}+\cos k_{y}-2\right)$ and $f_{l}=2(m_{l}\tau_{z}-c_{l})$. By performing
the Fourier transform $c_{\bm{k}}\rightarrow\sum_{n}c_{n}e^{-ik_{y}y_{n}}$, we
can discretize the tight-binding Hamiltonian as

$H=\sum_{n}c_{n}^{{\dagger}}h_{0}c_{n}+\sum_{n}\left(c_{n}^{{\dagger}}h_{y}c_{n+1}+h.c.\right),$
(12)

where the hopping matrices are given by

$\displaystyle h_{0}$ $\displaystyle=f_{0}+f_{1}\cos k_{z}+f_{2}\cos
k_{x}+f_{x}\sin k_{x}$
$\displaystyle-\alpha\sigma_{x}\tau_{x}\left[\sin\left(k_{z}+k_{x}\right)+\sin\left(k_{z}-k_{x}\right)\right]$
(13) $\displaystyle h_{y}$
$\displaystyle=\frac{f_{2}+if_{y}}{2}+\alpha\sigma_{x}\tau_{x}\sin k_{z}$
$\displaystyle+i\alpha\sigma_{y}\tau_{x}\frac{\cos\left(k_{z}+k_{x}\right)-\cos\left(k_{z}-k_{x}\right)}{2}$
(14)

with $f_{0}=-2f_{2}-f_{1}\cos k_{w}$ and
$f_{x(y)}=\lambda\sigma_{z(0)}\tau_{x(y)}$.

In Fig. 1, we plot the numerical spectrum and $\bm{k}$-resolved DOSs for Eq.
(12). As shown by the green dotted curves in Figs. 1(a)-(d), the self-
consistent calculation of Eq. (9) coincides with the results by the numerical
diagonalization of Eq. (12). As illustrated by Figs. 1(a) and (e), the surface
spectrums without $\varepsilon_{\bm{k}}$ and $\Lambda(\bm{k})$ are
$k_{z}$-independent and cross at $E_{D}$ with a quadruply degenerate line
connecting the two Dirac points. The flat Fermi line can be easily modified by
perturbations. This explains why the particle-hole symmetry and the momentum-
dependent gap term are important to the surface states. A nonzero
$\varepsilon_{\bm{k}}$ can not gap the surface spectrum, but will bend the
Fermi lines to a parabola, as shown by Fig. 1(b). As a result, the Fermi arcs
with opposite spin, due to the spin-dependent term in Eq. (8), will curve in
opposite direction at the Fermi level, forming a closed loop with a
discontinuous kink at the Dirac points, as indicated by Fig. 1(f).

By contrast, when the momentum-dependent gap term is included, the Fermi lines
with opposite spin, because of the noncommutation between $\sigma_{z}$ and
$\mathcal{H}_{\mathrm{surf}}^{\beta}\left(k_{x},k_{z}\right)$, will repel each
other and so remove the spin degeneracy. Since $\Lambda(\bm{k})=0$ for
$k_{\parallel}=0$ or $k_{z}=0$, the surface spectrums keep crossing at
$\bm{k}=0,\bm{K}_{\pm}$, as shown by Fig. 1(c). Therefore, a 2D Dirac point
develops on each surface, which coexists with the bulk Dirac points. The
surface Dirac points, similar to the bulk Dirac points, are robust against
$\Lambda(\bm{k})$, but, differently, the energy location of the surface Dirac
points can be modulated by the particle-hole asymmetry. In the presence of
particle-hole symmetry, i.e., $\varepsilon_{\bm{k}}=0$, the energy location of
the surface Dirac points is identical to the bulk Dirac points, such that at
$E_{F}=E_{D}$, the Fermi line in Fig. 1(e) deforms into three Fermi points in
Fig. 1(g). When the particle-hole symmetry is broken by a finite
$\varepsilon_{\bm{k}}$, the surface Dirac points will shift away from $E_{D}$,
after that the surface Fermi point around $\bm{k}=0$ turns to a Fermi loop, as
demonstrated by Fig. 1(h), which prevents the Fermi arcs from connecting the
bulk Dirac points.

Consequently, although the surface spectrums remain intersecting with the bulk
Dirac points, the surface states will experience a Lifshitz transition with
the Fermi arcs changing from a curve to some discrete gap-closing points and
further to a Fermi loop coexisting with the bulk Dirac points, during which
process the Weyl orbits can be turned off without destroying the topological
properties of the bulk energy band.

Figure 3: The Hall conductivity (left axis) and DOS (right axis) for a given
$k_{z}$ with (a) $\alpha=0$, (b) $\alpha=0.1$ and (c) $\alpha=0.3$. (d) The
Fermi surface for $E_{F}=0$ and $\alpha=0.3$ in the absence of magnetic field.
The parameters are the same as Fig. 2.

##  IV LLs and probability density distribution

Upon application of an external magnetic field, a Peierls phase should be
added to the hopping matrices as electrons jump from position $\bm{r}_{n}$ to
$\bm{r}_{m}$, i.e., $t_{mn}\rightarrow t_{mn}e^{-i\varphi_{mn}}$ with
$\varphi_{mn}=\frac{2\pi}{\Phi_{0}}\int_{\bm{r}_{m}}^{\bm{r}_{n}}\bm{A}\cdot
d\bm{l}$ and $\Phi_{0}=h/e$ being the unit flux quantum. For the magnetic
field is applied in the $y$ direction, we can choose the Landau gauge
$\bm{A}=(0,0,-Bx)$. To include the magnetic field, we further discretize Eq.
(12) in direction $x$ as

$\displaystyle H$
$\displaystyle=\sum_{mn}c_{m,n}^{{\dagger}}(T_{m}c_{m,n}+T_{m,x}c_{m+1,n}+T_{m,y}c_{m,n+1}$
$\displaystyle+T_{m,\alpha}c_{m+1,n-1}-T_{m,\alpha}c_{m+1,n+1})+h.c.,$ (15)

in which the Fourier transform
$c_{n}\rightarrow\sum_{m}c_{m,n}e^{-ik_{x}x_{m}}$ has been adopted and the
hopping matrices are

$\displaystyle T_{m}$
$\displaystyle=\frac{f_{0}+f_{1}\cos\left(k_{z}-m/\ell_{B}^{2}\right)}{2},$
(16) $\displaystyle T_{m,\alpha}$
$\displaystyle=\frac{\alpha}{2}\sigma_{y}\tau_{x}\sin\left(k_{z}-\frac{m+1/2}{\ell_{B}^{2}}\right),$
(17) $\displaystyle T_{m,y}$
$\displaystyle=\frac{f_{2}+if_{y}}{2}+\alpha\sigma_{x}\tau_{x}\sin\left(k_{z}-m/\ell_{B}^{2}\right),$
(18) $\displaystyle T_{m,x}$
$\displaystyle=\frac{f_{2}-if_{x}}{2}-\alpha\sigma_{y}\tau_{x}\sin\left(k_{z}-\frac{m+1/2}{\ell_{B}^{2}}\right),$
(19)

with $\ell_{B}=\sqrt{\hbar/\left(eB\right)}$ being the magnetic length.

After introducing the magnetic field, we continue to demonstrate the relation
between Eq. (11) and Eq. (15). By the inverse Fourier transform on Eq. (15),
we can arrive at

$\displaystyle\mathcal{\tilde{H}}_{\bm{k}}$
$\displaystyle=f_{1}\left(\cos\kappa_{z}-\cos k_{w}\right)+f_{2}\left(\cos
k_{x}+\cos k_{y}-2\right)$ $\displaystyle+\lambda\left(\sin
k_{x}\tau_{x}\sigma_{z}-\sin
k_{y}\tau_{y}\right)-\alpha\tau_{x}[(\frac{\sin\kappa_{z}^{+}-\sin\kappa_{z}^{-}}{2}$
$\displaystyle-\cos k_{y}\sin\kappa_{z})\sigma_{x}-\sin
k_{y}\frac{\cos\kappa_{z}^{+}-\cos\kappa_{z}^{-}}{2}\sigma_{y}]$ (20)

with $\kappa_{z}=k_{z}-x/\ell_{B}^{2}$ and
$\kappa_{z}^{\pm}=\pm\kappa_{z}+k_{x}$. Here, we have used the Baker-Campbell-
Hausdorff formula, i.e.,
$e^{\hat{A}+\hat{B}}=e^{\hat{A}}e^{\hat{B}}e^{-\frac{1}{2}[\hat{A},\hat{B}]}$.
It is clear that
$\mathcal{\tilde{H}}_{\bm{k}}=\mathcal{H}_{\bm{k}\rightarrow\bm{k}+e\bm{A}/\hbar}^{\mathrm{tb}}$
follows the Peierls substitution. However, for $\alpha\neq 0$, one should be
very careful when using the direct Peierls substitution, because of the cross
momentum term in $\Lambda(\bm{k})$. For example, as we transform Eq. (20) to
the lattice form for numerical calculation, there will emerge an additional
phase $\ell_{B}^{-2}/2$, such as in Eqs. (17) and (19). The underlying physics
for the additional phase is that after the Peierls substitution, the different
momentum components can be noncommutative, e.g., $[k_{x},\kappa_{z}]=$
$i\ell_{B}^{-2}$, and the Baker-Campbell-Hausdorff formula must be adopted.
For instance, when transforming Eq. (12) to Eq. (15), one should express the
trigonometric functions in the exponential form and then introduce the Peierls
phase via the transforms $e^{ik_{z}}\rightarrow e^{i\kappa_{z}}$ and
$e^{i\left(k_{x}\pm k_{z}\right)}\rightarrow e^{i\kappa_{z}^{\pm}}=e^{\pm
i\left(\kappa_{z}-\ell_{B}^{-2}/2\right)}e^{ik_{x}}$.

By diagonalizing Hamiltonian (15) numerically, we can obtain the LLs and
spatial distribution of the electron probability density, as displayed in Fig.
2. Through the spatial distribution of the probability density, we can tell
easily whether or not the LLs are formed by the Weyl orbits. From Figs. 2(a)
and (e), we see that the LLs can be formed from the Weyl-orbit mechanism, even
when there is no curvature in the Fermi arcs. In the Weyl-orbit mechanism, the
surface fermions, driven by the magnetic field, will move along the Fermi arcs
from one Dirac valley to the other and tunnel to the opposite surface at the
Dirac points via the bulk states. Therefore, the probability density exhibits
a closed loop with two bright stripes crossing the bulk and connecting the
surface states, as seen from Fig. 2(e). The width of the bright stripes $\sim
2\ell_{B}$ relates to the cyclotron radius of the bulk Dirac fermions and the
distance between the stripes encodes the momentum distance between the Dirac
points. The cyclotron center $x_{c}=\ell_{B}^{2}k_{z}$ of the fermions in
different Dirac valleys differs by $\Delta x_{c}=2\ell_{B}^{2}k_{w}$, which is
exactly the distance between the two bright stripes.

A finite $\varepsilon_{\bm{k}}$ that curves the Fermi arcs in the surface
Brillouin zone will shift the LLs, integrally, by modifying the magnetic flux
enclosed by the Weyl orbits, as demonstrated by Figs. 2(b) and (f). In this
case, the Weyl orbits will not be destroyed, so that the LLs remain evenly
spaced. As expected, the LLs will respond dramatically to $\Lambda(\bm{k})$,
see Figs. 2(c) and (d) where the LLs distribute irregularly, for the Weyl-
orbit mechanism can be turned off by the surface Dirac cone. The probability
densities displayed in Figs. 2(g) and (h), as well as Fig. 4(c), illustrate
that the bulk and surface states can support the cyclotron orbit and form the
LLs, independently. The superposition of the bulk and surface LLs results in
the complicated LLs in Figs. 2(c) and (d).

Figure 4: (a)-(b) Evolution of the Hall conductivity with the sample
thickness, i.e., $N_{y}=(30,40,50)$, for (a) $\alpha=0$ with the inset showing
that the LLs are from the Weyl-orbit mechanism, and (b) $\alpha=0.3$ with the
red lines marked by $A$-$D$ denoting the LLs for $N_{y}=40$. (c) The
probability density of the LLs marked in (b), which indicates that the surface
and bulk states are disconnected, i.e., the Weyl orbits have been turned off.
The parameters are chosen the same as Fig. 3.

##  V Evolution of the QHE with the surface Lifshitz transition

As discussed above, in the presence of the momentum-dependent gap term, the
Weyl orbit near the Dirac points will vanish. However, the LLs remain
discrete, implying that the QHE in topological DSMs can happen without
involving the Weyl-orbit mechanism. As $k_{z}$ is a good quantum number, we
can calculate the Hall conductivity from the Kubo formulaWang  _et al._
(2017); Li  _et al._ (2020); Ma  _et al._ (2021); Geng  _et al._ (2021); Chang
_et al._ (2021)

$\displaystyle\sigma_{xz}$ $\displaystyle=\frac{i\hbar
e^{2}}{L_{x}L_{z}}\sum_{k_{z},\mu\neq\nu}\left(f_{\nu,k_{z}}-f_{\mu,k_{z}}\right)$
$\displaystyle\times\frac{\langle\psi_{k_{z},\mu}|\hat{\upsilon}_{x}|\psi_{k_{z},\nu}\rangle\langle\psi_{k_{z},\nu}|\hat{\upsilon}_{z}|\psi_{k_{z},\mu}\rangle}{\left(\varepsilon_{k_{z},\mu}-\varepsilon_{k_{z},\nu}\right)^{2}},$
(21)

where $\varepsilon_{\mu,k_{z}}$ denote respectively the $\mu$-th energy of Eq.
(15), and the velocity operators are defined as
$\hat{\upsilon}_{x}=i\hbar^{-1}\sum_{mn}\left[H,c_{m,n}^{{\dagger}}x_{m}c_{m,n}\right]$
and $\hat{\upsilon}_{z}=\partial H/(\hbar\partial k_{z})$. Here,
$f_{\mu,k_{z}}=[1+\exp(\frac{\varepsilon_{\mu,k_{z}}-E_{F}}{k_{B}T})]^{-1}$ is
the Fermi-Dirac distribution function, with $k_{B}$ and $T$ being the
Boltzmann’s constant and temperature, respectively.

In Fig. 3, we present the numerical results for the Hall conductivity near the
bulk Dirac points. As can be seen, the Hall conductivity exhibits a step-wise
structure and jumps whenever the Fermi level goes across a LL. The Hall
plateaus for $\alpha=0$ are quantized as $2ne^{2}/h$, with $n$ as an integer
and the factor $2$ owing to the spin degeneracy of the LLs. The spin
degeneracy of the LLs can be characterized by the $k_{z}$-resolved DOS plotted
in the right axis of Fig. 3. In the absence of the momentum-dependent gap
term, the LLs are evenly spaced and therefore, the widths of the Hall plateaus
are identical, as shown by Fig. 3(a). With increasing $\alpha$, the Weyl
orbits near the bulk Dirac points will be broken gradually, and as a result,
the LLs become complicated and the widths of the Hall plateaus turn to be
irregular, as observed in Figs. 3(b) and (c). With the Weyl orbits destroyed,
the spin degeneracy of the LLs will be removed, leading to the emergence of
odd Hall plateaus in Figs. 3(b) and (c).

The thickness dependence of the quantized Hall plateaus is one of the
characteristic signals for the 3D QHE Zhang  _et al._ (2019). The Weyl-orbit-
mediated 3D QHE is thickness-dependent, as shown by Fig. 4(a), where the Hall
plateaus increase synchronously in width as $N_{y}$ decreases. However, from
Fig. 4(b), we see that when the Weyl orbits have been turned off by the
surface Dirac cone, the Hall plateaus can still be sensitive to the thickness
of the sample. The thickness-dependent QHE in the absence of the Weyl orbits
can be attributable to the bulk LLs, as a result of the energy quantization by
the confinement in direction $y$, as illustrated by Fig. 4(c), where the
probability density exhibits a standing wave configuration in analogous to the
one for the infinite well. In contrast to the bulk LLs, the surface LLs are
less sensitive to the thickness, as shown by the LL marked by $C$ in Fig.
4(b). While the surface LLs are thickness-independent, the Hall plateau close
to them can display thickness dependence for the surface LLs are surrounded by
the bulk LLs.

##  VI Conclusion

In summary, we have investigated the QHE in topological DSMs under
consideration of the surface Lifshitz transition modulated by a bulk momentum-
dependent gap term. It is found that the bulk momentum-dependent gap term, as
a higher-order momentum term, can be neglected in study of the bulk
topological properties, but it will dramatically deform the Fermi arcs and
lead to the surface Lifshitz transition. During the surface Lifshitz
transition, a 2D Dirac cone will develop for the surface states, which keeps
the Fermi arcs from connecting the bulk Dirac points and, as a result, the
Weyl orbits can be turned off without breaking the topology of the bulk energy
band. In response to the Lifshitz transition, the 3D QHE because of the Weyl
orbits will break down, along with that the quantized Hall plateaus turn to be
irregular. As a bulk perturbation, the momentum-dependent gap term is quite
common in real materials. Therefore, in addition to the bulk Weyl nodes and
surface Fermi arcs, the Lifshitz transition of the surface states also plays
an important part in realization of stable Weyl orbits.

##  VII acknowledgements

This work was supported by the National NSF of China under Grants No.
12274146, No. 11874016, No. 12174121 and No. 12104167, the Guangdong Basic and
Applied Basic Research Foundation under Grant No. 2023B1515020050, the Science
and Technology Program of Guangzhou under Grant No. 2019050001 and the
Guangdong Provincial Key Laboratory under Grant No. 2020B1212060066.

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ampmtime

# Nowhere dense Ramsey Sets

Vojtěch Rödl  and  Marcelo Sales  Department of Mathematics, Emory University,
Atlanta, GA, USA<EMAIL_ADDRESS>Department of Mathematics, University of
California, Irvine, CA, USA<EMAIL_ADDRESS>
###### Abstract.

A set of points $S$ in Euclidean space $\mathbb{R}^{d}$ is called Ramsey if
any finite partition of $\mathbb{R}^{\infty}$ yields a monochromatic copy of
$S$. While characterization of Ramsey set remains a major open problem in the
area, a stronger “density” concept was considered in [FR90]: If $S$ is a
$d$-dimensional simplex, then for any $\mu>0$ there is an integer
$d:=d(S,\mu)$ and finite configuration $X\subseteq\mathbb{R}^{d}$ such that
any subconfiguration $Y\subseteq X$ with $|Y|\geqslant\mu|X|$ contains a copy
of $S$. Complementing this, here we show the existence of $\mu:=\mu(S)$ and of
an infinite configuration $X\subseteq\mathbb{R}^{\infty}$ with the property
that any finite coloring of $X$ yields a monochromatic copy of $S$, yet for
any finite set of points $Y\subseteq X$ contains a subset $Z\subseteq Y$ of
size $|Z|\geqslant\mu|Y|$ without a copy of $S$.

The authors were supported by NSF grant DMS 1764385. The first author was also
supported by NSF grant DMS 2300347 and the second author by US Air Force grant
FA9550-23-1-0298.

##  1\. Introduction

We will find convenient to present our discussion in the framework of
$\mathbb{R}^{\infty}$, by which we understand a subspace of $\ell^{2}$
consisting of infinite sequences of real numbers with all but finitely many
nonzero entries and with $\mathbb{R}^{\infty}$ equipped by the usual euclidean
metric. In other words, we can view $\mathbb{R}^{\infty}$ as the infinite
union $\mathbb{R}^{\infty}=\bigcup_{d=1}^{\infty}\mathbb{R}^{d}$, where we
understand that the copies of $\mathbb{R}^{d}$ are being included in one
another.

For two configuration of points $A,B\subseteq\mathbb{R}^{\infty}$ we will
write

$\displaystyle A\rightarrow(B)_{r}$

to denote the fact that any $r$-coloring of $A$ yields a monochromatic copy of
$B$. By a (congruent) copy of $B$, we mean a subconfiguration
$B^{\prime}\subseteq A$ that is isometric to $B$, i.e., that exists a
bijective map $\varphi:B\rightarrow B^{\prime}$ such that

$\displaystyle||b_{1}-b_{2}||=||\varphi(b_{1})-\varphi(b_{2})||$

for every $b_{1},b_{2}\in B$. Given two configurations $A,B$ we say that $B$
is contained in $A$, and denote $B\subseteq A$, if there exists a copy
$A^{\prime}$ of $A$ such that $B\subseteq A^{\prime}$ (in the set theoretical
sense).

A finite configuration $S$ is said to be Ramsey if
$\mathbb{R}^{\infty}\rightarrow(S)_{r}$ for every integer $r\geqslant 1$. The
concept was introduced in [EGMRSS73] by Erdős, Graham, Montgomery, Rothschild,
Spencer and Strauss, who proved that the vertex set of every brick
(rectangular parallelepiped) of arbitrary finite dimension is Ramsey. The list
of Ramsey configurations was extended by a few more configurations in [FR90,
FR86, K91, K92]. On the other hand, the authors of [EGMRSS73] also proved that
any Ramsey set is spherical, i.e., all points of $S$ lie on some finite
dimensional sphere. They asked if the opposte implication is also true: If any
spherical set is Ramsey. In [Gr04] Ron Graham offered $\$1000$ dollars for
deciding if this implication holds as well. Based on the evidence coming from
known Ramsey configurations Leader, Russel and Walters [LRW12] proposed an
alternative conjecture. Calling a finite set transitive if its symmetry group
is transitive, i.e., if all points play the same role, their conjecture states
that Ramsey sets are precisely the transitive sets together with their
subsets.

While the progress on these conjectures was very small, some alternative
concepts were considered in [Gr83, Gr85, MR95, FR90, FPRR18, CF18]. In this
paper we will introduce another concept related to the Euclidean Ramsey
problems. A $d$-dimensional simplex $S$ is a configuration consisting of $d+1$
affinely independent points in $\mathbb{R}^{\infty}$. In [FR90] it was proved
that all simplices are Ramsey. One interesting feature of their proof is that
they actually show the following stronger statement.

######  Theorem 1.1 ([FR90]).

Let $S\subseteq\mathbb{R}^{\infty}$ be a $d$-dimensional simplex and $0<\mu<1$
real number. Then there exists finite configuration
$Y\subseteq\mathbb{R}^{\infty}$ such that any subconfiguration $Z\subseteq Y$
of size $|Z|\geqslant\mu|Y|$ contains a copy of $S$.

In other words, Theorem 1.1 not only finds a configuration $Y$ such that
$Y\rightarrow(S)_{r}$, but also with the extra property that any subset of
positive density contains a copy of $S$. One of the goals of this paper is to
show an alternative construction of the fact that simplices are Ramsey where
our set $Y$ does not have the density property. The following definition is
central for our exposition.

######  Definition 1.2.

A finite configuration $X\subseteq\mathbb{R}^{\infty}$ is called P-Ramsey if
there exists a configuration $Y\subseteq\mathbb{R}^{\infty}$ and a real number
$\mu>0$ such that the following holds:

  1. (i)

$Y\rightarrow(X)_{r}$ holds for every integer $r\geqslant 1$.

  2. (ii)

Every finite subconfiguration $Y^{\prime}\subseteq Y$ contains a configuration
$Z\subseteq Y^{\prime}$ with $|Z|\geqslant\mu|Y^{\prime}|$ such that $Z$ is
$X$-free

The definition is motivated by a problem formulated by Pisier in [G83] (see
also [ENR90, RRS22, NRS22]). Note that statement $(ii)$ of the P-Ramsey
definition is in contrast with the density statement introduced in Theorem
1.1, since it says that every finite subconfiguration contains a large set
without a copy of $X$.

Clearly, if $X$ is P-Ramsey, then $X$ is Ramsey. However, the converse is not
so clear. In this paper, we start the study of P-Ramsey configurations by
showing the following two results.

######  Theorem 1.3.

All simplices are P-Ramsey.

We say that a configuration $B\subseteq\mathbb{R}^{\infty}$ is a
$d$-dimensional brick if there exists positive real numbers
$a_{1},\ldots,a_{d}\in\mathbb{R}$ such that $B$ is congruent to the set

$\displaystyle\left\\{(x_{1},\ldots,x_{d}):\>x_{i}=0\text{ or
}x_{i}=a_{i},\,1\leqslant i\leqslant d\right\\}.$

######  Theorem 1.4.

All bricks are P-Ramsey.

The paper is organized as follows: In Section 2 we prove that segments are
P-Ramsey. We introduce our main technical result in Section 3, a variation of
the product theorem for P-Ramsey configurations, and prove Theorem 1.4.
Section 4 is devoted to the proof of Theorem 1.3.

##  2\. Segments are P-Ramsey

We prove in this section that segments are P-Ramsey. In fact, we will prove a
sligthly stronger statement. Recall that a weight vector
$\mathbf{w}:X\rightarrow[0,1]$ is stochastical if $\sum_{x\in
X}\mathbf{w}(x)=1$.

######  Lemma 2.1.

Let $A$ be a segment of distance $a$ and $\gamma>0$ be a real number. Then
there exists a countable configuration $Y_{A}\subseteq\mathbb{R}^{\infty}$
satisfying the following:

  1. (i)

The set of squares of all distances of points in $Y_{A}$ is

$\displaystyle\left\\{a^{2},\frac{a^{2}}{1+\gamma+\gamma^{2}},\frac{(1+\gamma^{2})a^{2}}{1+\gamma+\gamma^{2}},\frac{\gamma^{2}a^{2}}{1+\gamma+\gamma^{2}}\right\\}$

  2. (ii)

$Y_{A}\rightarrow(C)_{r}$ holds for every $r\geqslant 1$ and finite
configuration $C\subseteq Y_{A}$.

  3. (iii)

For every finite subconfiguration $Y^{\prime}\subseteq Y_{A}$ and stochastic
weight vector $\mathbf{w}:Y^{\prime}\rightarrow[0,1]$, there exists a
configuration $Z\subseteq Y^{\prime}$ with no segments of lenght $a$ such that
$\sum_{z\in Z}\mathbf{w}(z)\geqslant\frac{1}{4}$.

  4. (iv)

$Y_{A}$ does not contain an equilateral triangle of sides of lenght $a$.

The proof of Lemma 2.1 uses the following result about independent sets of a
shift graph. The shift graph $\operatorname{Sh}(2,\mathbb{N})$ is the graph
with vertex set $V(\operatorname{Sh}(2,\mathbb{N}))=\mathbb{N}^{(2)}$, i.e.,
the pairs of natural numbers, and edge set

$\displaystyle
E(\operatorname{Sh}(2,\mathbb{N}))=\left\\{\left\\{\\{x,y\\},\\{y,z\\}\right\\}:\>x<y<z\right\\}.$

######  Claim 2.2.

Let $\operatorname{Sh}(2,\mathbb{N})$ be the shift graph on the pairs of
$\mathbb{N}$. Then for every finite subset $X\subseteq
V(\operatorname{Sh}(2,\mathbb{N}))$ and stochastical weight vector
$\mathbf{w}:X\rightarrow[0,1]$, there exists an independent set $I\subseteq X$
such that

$\displaystyle\sum_{i\in I}\mathbf{w}(i)\geqslant\frac{1}{4}.$

###### Proof.

Let $X\subseteq\mathbb{N}^{(2)}$ be a finite subset of vertices of
$\operatorname{Sh}(2,\mathbb{N})$ and let $\mathbf{w}:X\rightarrow[0,1]$ be a
stochastic weight vector. Consider a random coloring
$c:\mathbb{N}\rightarrow\\{0,1\\}$, where each integer $n$ is colored
independently with probability

$\displaystyle\mathbb{P}(c(n)=0)=\frac{1}{2}.$

Let $X_{0,1}$ be the random set defined by

$\displaystyle X_{0,1}=\left\\{\\{x,y\\}\in X:\>x<y\text{ and
}c(x)=0,\,c(y)=1\right\\}.$

That is, $X_{0,1}$ are the ordered pairs of $X$ such that the first integer is
of color $0$ and the last one of color $1$. One can see that $X_{0,1}$ is an
independent set in $\operatorname{Sh}(2,\mathbb{N})$. Moreover, by letting

$\displaystyle Z_{0,1}=\sum_{x\in X_{0,1}}\mathbf{w}(x).$

we have that

$\displaystyle\mathbb{E}(Z_{0,1})=\sum_{\\{x,y\\}\in
X}\mathbb{P}\big{(}\\{c(x)=0\\}\wedge\\{c(y)=1\\}\big{)}\mathbf{w}(\\{x,y\\})=\sum_{\\{x,y\\}\in
X}\frac{1}{4}\mathbf{w}(\\{x,y\\})=\frac{1}{4}.$

Thus, by the first moment, with positive probability there is a coloring $c$
such that $X_{0,1}$ is an independent set satisfying the statement of the
claim. ∎

###### Proof of Lemma 2.1.

Let $\\{e_{i}\\}_{i=1}^{\infty}$ be the standard basis of
$\mathbb{R}^{\infty}$. We construct a configuration
$Y_{A}=\\{y_{e}\\}_{e\in\mathbb{N}^{(2)}}\subseteq\mathbb{R}^{\infty}$ by
associating to each pair $e=\\{i,j\\}\in\mathbb{N}^{(2)}$, $i<j$, the point

$\displaystyle y_{e}=\beta e_{i}-\beta\gamma e_{j},$

where $\beta=\frac{a}{\sqrt{2(1+\gamma+\gamma^{2})}}$. We claim that the
configuration $Y_{A}$ satisfies properties $(i)$, $(ii)$, $(iii)$ and $(iv)$
of Lemma 2.1.

Property $(i)$ comes from the fact that given two pairs $e=\\{i,j\\}$,
$e^{\prime}=\\{i^{\prime},j^{\prime}\\}\in\mathbb{N}^{(2)}$ the square of the
distance between $y_{e}$ and $y_{e^{\prime}}$ can assume the following values

$\displaystyle||y_{e}-y_{e^{\prime}}||^{2}=\begin{cases}2\beta^{2}\gamma^{2},\quad&\text{if
}i=i^{\prime}\\\ 2\beta^{2},\quad&\text{if }j=j^{\prime}\\\
2\beta^{2}(1+\gamma+\gamma^{2}),\quad&\text{if }i=j^{\prime}\text{ or
}i^{\prime}=j\\\ 2\beta^{2}(1+\gamma^{2}),\quad&\text{if
}\\{i,j\\}\cap\\{i^{\prime},j^{\prime}\\}=\emptyset\end{cases}$

By plugging $\beta=\frac{a}{\sqrt{2(1+\gamma+\gamma^{2})}}$ we obtain the set
of distances of the statement. Moreover, another important consequence of the
computation is that $||y_{e}-y_{e^{\prime}}||=a$ if and only if $e\sim
e^{\prime}$ in $\operatorname{Sh}(2,\mathbb{N})$.

In order to prove $(ii)$, consider a finite configuration $C\subseteq Y_{A}$.
Naturally $C$ can be written as $C=\\{y_{e}\\}_{e\in E}$ for some
$E\subseteq\mathbb{N}^{(2)}$. Since $E$ is finite, there exists an integer $n$
such that $E\subseteq[n]^{(2)}$. An $r$-coloring of $Y_{A}$ corresponds to an
$r$-coloring of $\mathbb{N}^{(2)}$. By Ramsey theorem [R29], there exists a
set $W\subseteq\mathbb{N}$ of size $n$ such that $W^{(2)}$ is monochromatic.
Hence, this configuration $C^{\prime}=\\{y_{e}\\}_{e\in W^{(2)}}$ is
monochromatic. This implies property $(i)$, since $C^{\prime}$ contains a copy
of $C$.

To check property $(iii)$, let $Y^{\prime}\subseteq Y_{A}$ be a finite
subconfiguration of $Y_{A}$. By our construction, this corresponds to a finite
set $X\subseteq V(\operatorname{Sh}(2,\mathbb{N}))$. Let
$\mathbf{w}^{\prime}:X\rightarrow[0,1]$ be the stochastic weight vector given
by $\mathbf{w}^{\prime}(x)=\mathbf{w}(y)$, where $y\in Y^{\prime}$ is the
corresponding point to $x\in X$. Claim 2.2 applied to the vector
$\mathbf{w}^{\prime}$ gives us an independent set $I\subseteq X$ in
$\operatorname{Sh}(2,\mathbb{N})$ such that $\sum_{i\in
I}\mathbf{w}^{\prime}(i)\geqslant\frac{1}{4}$. This corresponds to a
subconfiguration $Z\subseteq Y^{\prime}$ with no segments of length $a$ and
such that

$\displaystyle\sum_{z\in Z}\mathbf{w}(z)\geqslant\frac{1}{4}.$

Finally, property $(iv)$ follows from the fact that an equilateral triangle of
sides of length $a$ corresponds to a triangle in
$\operatorname{Sh}(2,\mathbb{N})$ and $\operatorname{Sh}(2,\mathbb{N})$ is
triangle free. ∎

##  3\. Robust configurations

One of the main techniques developed in [EGMRSS73] to prove that a
configuration is Ramsey is the product theorem

######  Theorem 3.1 ([EGMRSS73], Theorem 20).

Let $A$ and $B$ be finite configurations which are Ramsey and
$X,Y\subseteq\mathbb{R}^{\infty}$ be such that $X\rightarrow(A)_{r}$ and
$Y\rightarrow(B)_{r}$ for every $r\geqslant 1$. Then $X\times
Y\rightarrow(C)_{r}$ for $C\subseteq A\times B$ for every $r\geqslant 1$.

Unfortunately, it is not clear if a similar statement holds for P-Ramsey
configurations. The goal of this sections is to develop a partial version of
the product theorem that will enable us to prove Theorems 1.3 and 1.4.

######  Definition 3.2.

We say that a countable configuration $Y$ is robust if for every finite
configuration $C$ with $C\subseteq Y$ we have that $Y\rightarrow(C)_{r}$ for
every $r\geqslant 1$.

Note for instance, that by property $(i)$ of Lemma 2.1 we have the following.

######  Corollary 3.3.

Let $Y_{A}$ be the configuration obtained by Lemma 2.1. Then $Y_{A}$ is a
robust configuration.

The following is our main result in the section. Recall that by $B\subseteq A$
we understand that there exists $A^{\prime}$ copy of $A$ such that $B\subseteq
A$.

######  Theorem 3.4.

Let $B$ be a brick and $Y$ be a robust configuration. If $F\subseteq B\times
Y$ and $F\not\subseteq Y$, then $F$ is P-Ramsey.

Theorem 3.4 is a consequence of the following lemma.

######  Lemma 3.5.

Let $Y$ be a robust configuration, $A$ be a segment and $F$ a finite
configuration with $|F|>1$. Then the following holds:

  1. (a)

If $F\subseteq A\times Y$ and $F\not\subseteq Y$, then $F$ is P-Ramsey.

  2. (b)

If $F\not\subseteq A\times Y$, then there exists a robust configuration
$\tilde{Y}$ such that $A\times Y\subseteq\tilde{Y}$ and
$F\not\subseteq\tilde{Y}$.

###### Proof.

Let $a$ be the length of the segment $A$, also let $D_{Y}$ be the set of all
distances in $Y$ and let $D_{F}$ be the set of all distances in $F$. Consider
the field extension $L=\mathbb{Q}(a,D_{Y},D_{F})$ of $\mathbb{Q}$, where
$\mathbb{Q}(a,D_{Y},D_{F})$ is the minimal field containing $a$, $D_{Y}$,
$D_{F}$ and $\mathbb{Q}$. Since $D_{Y}\cup D_{F}\cup\\{a\\}$ is countable, we
have that $L$ is a countable extension of $\mathbb{Q}$ and consequently
$L\neq\mathbb{R}$. Let $\gamma\in\mathbb{R}$ be a transcedental number over
$L$, i.e.,

there is no polynomial $p\in L[x]$ such that $p(\gamma)=0$ (1)

Let $Y_{A}$ be the configuration obtained by Lemma 2.1 with parameters $a$ and
$\gamma$. By property $(iii)$ the set of all square distances is given by

$\displaystyle\left\\{a^{2},\,\frac{a^{2}}{1+\gamma+\gamma^{2}},\,\frac{(1+\gamma^{2})a^{2}}{1+\gamma+\gamma^{2}},\,\frac{\gamma^{2}a^{2}}{1+\gamma+\gamma^{2}}\right\\}$

Note that while $a^{2}\in L$, due to the fact that $\gamma$ is transcedental,
the other three distances are not in $L$. Indeed, to illustrate, assume for
example that $\frac{a^{2}}{1+\gamma+\gamma^{2}}\in L$. Then there exists $b\in
L$ such that

$\displaystyle\frac{a^{2}}{1+\gamma+\gamma^{2}}=b.$

This implies that $\gamma$ is a root of the polynomial $p\in L[x]$ given by
$p(x)=bx^{2}+bx+b-a^{2}$, which contradicts the assumption that $\gamma$ is
transcedental over $L$.

Before we address statements $(a)$ and $(b)$ of Lemma 3.5, we will prove the
following claim. Let $\pi_{A}:Y_{A}\times Y\rightarrow Y_{A}$ and
$\pi_{Y}:Y_{A}\times Y\rightarrow Y$ be the projection maps of $Y_{A}\times Y$
on $Y_{A}$ and $Y$, respectively.

######  Claim 3.6.

Let $F\subseteq Y_{A}\times Y$. Then either $F\subseteq Y$ or $\pi_{A}(F)$ is
a copy of $A$.

###### Proof.

If $\pi_{A}(F)$ is a single point, then $F\subseteq Y$ and there is nothing to
do. Thus, we may assume that $|\pi_{A}(F)|\geqslant 2$. Let $p,q$ be two
points of $F$ such that $p^{\prime}=\pi_{A}(p)$ and $q^{\prime}=\pi_{A}(q)$
are distinct. We claim that $||p^{\prime}-q^{\prime}||=a$. Let
$p^{\prime\prime}=\pi_{Y}(p)$ and $q^{\prime\prime}=\pi_{Y}(q)$. Since all
distances from points of $F$ and $Y$ are in $L$, we have that
$||p-q||^{2},||p^{\prime\prime}-q^{\prime\prime}||^{2}\in L$. Thus, by
Pythagoras theorem we have

$\displaystyle||p^{\prime}-q^{\prime}||^{2}=||p-q||^{2}-||p^{\prime\prime}-q^{\prime\prime}||^{2}\in
L.$ (2)

On the other hand, by Lemma 2.1 we have that

$\displaystyle||p^{\prime}-q^{\prime}||^{2}\in\left\\{a^{2},\,\frac{a^{2}}{1+\gamma+\gamma^{2}},\,\frac{(1+\gamma^{2})a^{2}}{1+\gamma+\gamma^{2}},\,\frac{\gamma^{2}a^{2}}{1+\gamma+\gamma^{2}}\right\\}.$

Due to our choice of $\gamma$, the value $a^{2}$ is the only one of the four
values in the field $L$. Hence, due to (2) we have
$||p^{\prime}-q^{\prime}||=a$.

Suppose that $|\pi_{A}(F)|\geqslant 3$. Then by the previous paragraph, there
is an equilateral triangle of sides of length $a$ in $Y_{A}$, which
contradicts property $(iv)$ of Lemma 2.1. Therefore, $\pi_{A}(F)$ is a segment
of length $a$. ∎

Now we prove statement $(a)$ of Lemma 3.5. Let $F$ be a finite configuration,
$|F|>1$, such that $F\subseteq A\times Y$ and $F\not\subseteq Y$ for a segment
$A$ and a robust configuration $Y$. By Corollary 3.3, the configuration
$Y_{A}$ is robust, wher $Y_{A}$ is defined with parameters $a$ and $\gamma$
satisfying (1. We claim that $Y_{A}\times Y$ testifies that $F$ is P-Ramsey.

To check property $(i)$ of Definition 1.2 we note that since $F\subseteq
A\times Y$, then there exists a finite configuration $C$ such that $F\subseteq
A\times C$. Because $Y$ is robust, then $Y\rightarrow(C)_{r}$ for every
$r\geqslant 1$. Lemma 2.1 gives us that $Y_{A}\rightarrow(A)_{r}$ for every
$r\geqslant 1$. Thus, by Theorem 3.1, we have that $Y_{A}\times
Y\rightarrow(F)_{r}$ for every $r\geqslant 1$.

In order to prove property $(ii)$ of Definition 1.2, let $V\subseteq
Y_{A}\times Y$ be a finite subconfiguration. Since $V$ is finite, there exists
a finite subconfiguration $X\subseteq Y_{A}$ such that $V\subseteq X\times Y$.
We partition $V$ into $V=\bigcup_{x\in X}V_{x}$ where $V_{x}=\pi_{A}^{-1}(x)$
are the elements of $V$ that projects to the point $x$ on $Y_{A}$. Let
$\mathbf{w}:X\rightarrow[0,1]$ be the stochastic weight vector defined by

$\displaystyle\mathbf{w}(x)=\frac{|V_{x}|}{|V|}.$

By property $(ii)$ of Lemma 2.1, there exists a subconfiguration $Z\subseteq
X$ with no segments of length $a$ such that

$\displaystyle\sum_{z\in Z}\mathbf{w}(z)\geqslant\frac{1}{4}.$ (3)

Consider the configuration $U=\bigcup_{z\in Z}V_{z}$. We claim that $U$ does
not contain a copy of $F$. Suppose to the contrary that $F\subseteq U$. Since
$F\not\subseteq Y$, then by Claim 3.6 the projection $\pi_{A}(U)$ contains a
segment of length $a$. However, $\pi_{A}(U)=Z$, which contains no segment of
length $a$, yielding a contradiction. Moreover, by (3)

$\displaystyle|U|=\sum_{z\in Z}|V_{z}|=\sum_{z\in
Z}|V|\mathbf{w}(z)\geqslant\frac{1}{4}|V|,$

which proves property $(ii)$ of Definition 1.2 with $\mu=\frac{1}{4}$. Hence,
$F$ is P-Ramsey.

Now we prove statement $(b)$ of Lemma 3.5. Suppose that $F\not\subseteq
A\times Y$. We claim that $\tilde{Y}=Y_{A}\times Y$ is a robust configuration
such that $F\not\subseteq Y_{A}\times Y$. We first show that $\tilde{Y}$ is
robust. If $C\subseteq\tilde{Y}$ is a finite configuration, then there exist
finite configurations $W_{A}\subseteq Y_{A}$ and $W\subseteq Y$ such that
$C\subseteq W_{A}\times W$. Since $Y_{A}$ and $Y$ are robust, we have that
$Y_{A}\rightarrow(W_{A})_{r}$ and $Y\rightarrow(W)_{r}$ for every $r\geqslant
1$. By Theorem 3.1, we obtain that $\tilde{Y}=Y_{A}\times
Y\rightarrow(C)_{r}$, which proves that $\tilde{Y}$ is robust.

Assume by contradiction that $F\subseteq Y_{A}\times Y$. Then by Claim 3.6, we
either have that $F\subseteq Y$ or $\pi_{A}(F)$ is a copy of $A$. In both
cases, we have that $F\subseteq A\times Y$, which contradicts the hypothesis.
∎

We are now able to prove Theorem 3.4.

###### Proof of Theorem 3.4.

Let $B$ be a $d$-dimensional brick and let $Y$ be a given robust
configuration. We will write $B=A_{1}\times\ldots\times A_{d}$ where $A_{i}$
is a segment. By the hypothesis of Theorem 3.4 we are also given $F$
satisfying $F\subseteq B\times Y$ and $F\not\subseteq Y$. Our goals is to
prove that $F$ is P-Ramsey. For that we will repeteadly apply Lemma 3.5. We
will construct a sequence $Y_{0},\ldots,Y_{\ell}$ of robust configurations
with the property that $F\not\subseteq Y_{i}$, for $0\leqslant
i\leqslant\ell$, as follows: Let $Y_{0}=Y$. Suppose that we already
constructed $Y_{0},\ldots,Y_{i}$. If $F\subseteq A_{i+1}\times Y_{i}$, then we
stop the process and set $\ell=i$. Otherwise, by statement $(b)$ of Lemma 3.5,
there exists a robust configuration $\tilde{Y}$ such that $A_{i+1}\times
Y_{i}\subseteq\tilde{Y}$ and $F\not\subseteq\tilde{Y}$. Set
$Y_{i+1}=\tilde{Y}$. A simple induction shows that for every $1\leqslant
i\leqslant\ell$

$\displaystyle A_{1}\times\ldots\times A_{i}\times Y\subseteq Y_{i}.$

Since $F\subseteq B\times Y=A_{1}\times\ldots\times A_{d}\times Y$, the
process terminates before the $d$-th step of the construction, i.e., $\ell<d$.
This implies that $F\subseteq A_{\ell+1}\times Y_{\ell}$ and $F\not\subseteq
Y_{\ell}$ and by statement $(a)$ of Lemma 3.5, we have that $F$ is P-Ramsey. ∎

A corollary of Theorem 3.4 is that bricks are P-Ramsey. In fact, we prove the
slighter stronger statement that in particular implies Theorem 1.4.

######  Corollary 3.7.

Let $B$ be a brick and $F\subseteq B$ be a subconfiguration with $|F|>1$. Then
$F$ is P-Ramsey.

###### Proof.

Suppose that $B$ is $d$-dimensional brick and write $B=A_{1}\times\ldots\times
A_{d}$, where $A_{i}$ is a segment of length $a_{i}$ and
$a_{1}\geqslant\ldots\geqslant a_{d}$. Let $\gamma>0$ be an arbitrary real
number and let $Y_{A_{d}}$ be the configuration obtained by Lemma 2.1 with
parameters $\gamma$ and $a_{d}$. Suppose that $F\subseteq Y_{A_{d}}$. By the
minimality of the segment $A_{d}$, we have that the minimum distance between
two points in $F$ is at least $a_{d}$. Moreover, by property $(iii)$ of Lemma
2.1, the diameter of $Y_{A_{d}}$ is exactly $a_{d}$. Hence, any two points of
$F$ has distance $a_{d}$. If $|F|\geqslant 3$, then $Y_{A_{d}}$ contains an
equilateral triangle of sides $a_{d}$. This contradicts property $(iv)$ of
Lemma 2.1. Thus, $F$ is a copy of the segment $A_{d}$ and in this case $F$ is
P-Ramsey by property $(ii)$ and $(iii)$ of Lemma 2.1.

Now suppose that $F\not\subseteq Y_{A_{d}}$. Since $A_{d}\subseteq Y_{A_{d}}$,
then $F\subseteq A_{1}\times\ldots\times A_{d}\subseteq
A_{1}\times\ldots\times A_{d-1}\times Y_{A_{d}}$. Therefore, $F$ satisfies the
hypothesis of Theorem 3.4 and we obtain that $F$ is P-Ramsey. ∎

##  4\. Simplices are P-Ramsey

In this section we prove Theorem 1.3. The proof follows the ideas from [FR90,
MR95]. First, we will introduce the terminology and auxiliary results from
those papers. The main idea will be to prove that any simplex $S$ can be
embedded in a product $B\times Y$, where $B$ is a brick and $Y$ is a robust
configuration.

To address the robust configuration consider the following definition: Let
$\\{e_{i}\\}_{i\geqslant 1}$ be the standard basis of $\mathbb{R}^{\infty}$.
Given an integer $k$, a vector $c=(c_{1},\ldots,c_{k})\in\mathbb{R}^{k}$ and a
$k$-tuple $J=(j_{1},\ldots,j_{k})\in\mathbb{N}^{(k)}$, we define the point
$\operatorname{spread}(c,J)\in\mathbb{R}^{\infty}$ as

$\displaystyle\operatorname{spread}(c,J)=\sum_{\ell=1}^{k}c_{\ell}e_{j_{\ell}}$

Given a subset of integers $A\subseteq\mathbb{N}$, one can then define the
configuration $\operatorname{Spread}(c,A)$ as

$\displaystyle\operatorname{Spread}(c,A)=\\{\operatorname{spread}(c,J):\>J\in
A^{(k)}\\}.$

The reason why spread configurations are interesting for us is twofold: One is
that those configurations approximate simplices very well. The second is that
it fits well in the context of P-Ramseyeness (see Claim 4.3 below) The next
result was proven in [MR95]. Given real number $\rho>0$, we denote by
$S_{\rho}(\mathbb{R}^{\infty})$ the sphere of radius $\rho$ in
$\mathbb{R}^{\infty}$. For a linear subspace $Z\subseteq\mathbb{R}^{\infty}$,
let $S_{\rho}(Z)=S_{\rho}(\mathbb{R}^{\infty})\cap Z$.

######  Proposition 4.1 ([MR95]).

For every $\delta>0$ and every integer $m$, there exist integer $n$, $k$, a
$k$-dimensional vector $c=(c_{1},\ldots,c_{k})\in\mathbb{R}^{\infty}$ with
$||c||=\rho$ and an $m$-dimensional subspace $Z\subseteq\mathbb{R}^{\infty}$
such that the following holds. For every $z\in Z$, there is a point
$y\in\operatorname{Spread}(c,[n])$ such that $||z-y||<\delta$.

Since any $d$-dimensional simplex can be embedded in any $d$-dimensional
vector space, we obtain the following corollary from Proposition 4.1.

######  Corollary 4.2.

For $\delta<\rho/2$ and $d$-dimensional simplex $S=\\{y_{0},\ldots,y_{d}\\}$
of circumradius $\rho(S)=\rho$, there exist integers $n$, $k$, a
$k$-dimensional vector $c=(c_{1},\ldots,c_{k})\in\mathbb{R}^{k}$ with
$||c||=\rho$ and a $d$-dimensional simplex
$S^{\prime}=\\{z_{0},\ldots,z_{d}\\}\subseteq\operatorname{Spread}(c,[n])$
such that $||y_{i}-z_{i}||<\delta$ for $0\leqslant i\leqslant d$.

The second reason is that Spread configurations are robust.

######  Claim 4.3.

$\operatorname{Spread}(c,\mathbb{N})$ is robust.

###### Proof.

Let $X\subseteq\operatorname{Spread}(c,\mathbb{N})$ be a finite configuration.
Then there exist $N$ and
$\mathcal{J}=\\{J_{1},\ldots,J_{t}\\}\subseteq[N]^{(k)}$ such that
$X=\\{\operatorname{spread}(c,J):J\in\mathcal{J}\\}$. Note that there exists a
bijective map $\varphi$ from $\operatorname{Spread}(c,\mathbb{N})$ to
$\mathbb{N}^{(k)}$ given by $\varphi(\operatorname{spread}(c,J))=J$. Thus, for
any finite coloring of $\operatorname{Spread}(c,\mathbb{N})$ there is a
corresponding coloring of $\mathbb{N}^{(k)}$. By Ramsey’s theorem, there exist
set $A\subseteq\mathbb{N}$ of size $N$ such that $A^{(k)}$ is monochromatic.
Therefore, the configuration $\operatorname{Spread}(c,A)$ is monochromatic.
The result follows now since $X\subseteq\operatorname{Spread}(c,A)$. ∎

Another important result for our proof is the next characterization of
configurations of points in an Euclidean space. Let $M=(m_{ij})_{0\leqslant
i,j\leqslant d}$ be a symmetric real matrix with zero entries on the main
diagonal. We say that the matrix $M$ is of negative type if

$\displaystyle\sum_{0\leqslant i<j\leqslant
d}m_{ij}\lambda_{i}\lambda_{j}\leqslant 0$ (4)

holds for all choices of $\lambda_{0},\ldots,\lambda_{d}$ with
$\lambda_{0}+\ldots+\lambda_{d}=0$ and
$\lambda_{0}^{2}+\ldots+\lambda_{d}^{2}=1.$

######  Theorem 4.4 ([S38]).

A finite configuration $X=\\{x_{0},\ldots,x_{d}\\}$ with distances
$d_{ij}=||x_{i}-x_{j}||$ can be embedded in the Euclidean space
$\mathbb{R}^{d}$ if and only if the matrix $M=(m_{ij})_{0\leqslant
i,j\leqslant d}$ given by $m_{ij}=d_{ij}^{2}$ is of negative type. Moreover,
$X$ is a $d$-dimensional simplex if and only if the inequality in (4) is
strict for all choices of $\lambda_{0},\ldots,\lambda_{d}$.

As a consequence of Theorem 4.4, we can show that all almost regular simplices
are realizable.

######  Corollary 4.5.

Let $d$ be an integer and $\beta,\varepsilon>0$ be real numbers such that
$\varepsilon<\frac{\beta}{64d^{2}}$. For any symmetric matrix of distances
$D=\\{d_{ij}\\}_{0\leqslant i,j\leqslant d}$ satisfying

$\displaystyle\beta-\varepsilon\leqslant d_{ij}\leqslant\beta+\varepsilon,$
(5)

there exists a $d$-dimensional simplex $S=\\{x_{0},\ldots,x_{d}\\}$ such that
$||x_{i}-x_{j}||=d_{ij}$ for every $0\leqslant i<j\leqslant d$.

###### Proof.

Let $M=(m_{ij})_{0\leqslant i,j\leqslant d}$ be the symmetric matrix with zero
entries in the main diagonal given by $m_{ij}=d_{ij}^{2}$ for $i\neq j$. For
real numbers $\lambda_{0},\ldots,\lambda_{d}$ satisfying
$\lambda_{0}+\ldots+\lambda_{d}=0$ and
$\lambda_{0}^{2}+\ldots+\lambda_{d}^{2}=1$ we have that

$\displaystyle
0=\left(\sum_{i=0}^{d}\lambda_{i}\right)^{2}=1+2\sum_{0\leqslant i<j\leqslant
d}\lambda_{i}\lambda_{j}.$

Hence,

$\displaystyle\sum_{0\leqslant i<j\leqslant
d}\lambda_{i}\lambda_{j}=-\frac{1}{2}$ (6)

Thus, by (5) and (6) we have

$\displaystyle\left|\left|\sum_{0\leqslant i<j\leqslant
d}m_{ij}\lambda_{i}\lambda_{j}+\frac{\beta^{2}}{2}\right|\right|=\left|\left|\sum_{0\leqslant
i<j\leqslant d}(m_{ij}-\beta^{2})\lambda_{i}\lambda_{j}\right|\right|$
$\displaystyle\leqslant\sum_{0\leqslant i<j\leqslant d}||m_{ij}-\beta^{2}||$
$\displaystyle\leqslant(d+1)^{2}(2\varepsilon\beta+\varepsilon^{2})<\frac{\beta^{2}}{4}.$

This implies that $\sum_{0\leqslant i<j\leqslant
d}m_{ij}\lambda_{i}\lambda_{j}<-\frac{\beta^{2}}{4}$ and $M$ is strictly of
negative type. Therefore, by Theorem 4.4 there exists a simplex
$S=\\{x_{0},\ldots,x_{d}\\}$ with $||x_{i}-x_{j}||=d_{ij}$ for $0\leqslant
i<j\leqslant d$. ∎

Finally, the last auxiliary result shows that any almost regular simplex can
be embedded in a brick.

######  Theorem 4.6 ([FR90, MR95]).

For every $\beta,d>0$, there exists a real number $\eta:=\eta(\beta,d)$ such
that the following holds. For any simplex $S=\\{w_{0},\ldots,w_{d}\\}$
satisfying

$\displaystyle\beta-\eta\leqslant||w_{i}-w_{j}||\leqslant\beta+\eta$

for $0\leqslant i<j\leqslant d$, there exists a $\binom{d+1}{2}$-dimensional
brick $B$ with $S\subseteq B$.

We are now ready to prove Theorem 1.3.

###### Proof of Theorem 1.3.

To prove that a simplex is P-Ramsey we will apply again Theorem 3.4, now
combined with ideas from [FR90, MR95]. We find convenient to divide the proof
in the next four steps:

Step 1: For a simplex $S=\\{x_{0},\ldots,x_{d}\\}$ with circumradius
$\rho(S)=\rho$, we will find a simplex $S_{1}=\\{y_{0},\ldots,y_{d}\\}$ a
small positive real number $\beta$ such that

$\displaystyle||y_{i}-y_{j}||^{2}=||x_{i}-x_{j}||^{2}-\beta$

for all $0\leqslant i\neq j\leqslant d$. This implies that the circumradius
$\rho(S_{1})=\rho^{\prime}<\rho$.

Let $M=(m_{ij})_{0\leqslant i,j\leqslant d}$ be the matrix given by
$m_{ij}=||x_{i}-x_{j}||^{2}$. Since $S$ is a simplex, by Theorem 4.4 there
exists $\gamma>0$ such that

$\displaystyle\sum_{0\leqslant i<j\leqslant
d}m_{ij}\lambda_{i}\lambda_{j}\leqslant-\gamma$

for all choices of $\lambda_{0},\ldots,\lambda_{d}$ with
$\lambda_{0}+\ldots+\lambda_{d}=0$ and
$\lambda_{0}^{2}+\ldots+\lambda_{d}^{2}=1$. Set $\beta=\frac{\gamma}{8d^{2}}$
and let $M^{\prime}=(m^{\prime}_{ij})_{0\leqslant i,j\leqslant d}$ be the
matrix defined by $m^{\prime}_{ij}=m_{ij}-\beta$ for $i\neq j$ and zero
entries in the main diagonal. Since $\beta(d+1)^{2}\leqslant 4\beta
d^{2}<\gamma/2$, then

$\displaystyle\sum_{0\leqslant i<j\leqslant
d}m^{\prime}_{ij}\lambda_{i}\lambda_{j}\leqslant-\beta\sum_{0\leqslant
i<j\leqslant
d}\lambda_{i}\lambda_{j}-\gamma\leqslant\beta(d+1)^{2}-\gamma<-\frac{\gamma}{2}<0.$

Consequently, $M^{\prime}$ is strictly negative, which implies that there
exists simplex $S_{!}=\\{y_{0},\ldots,y_{d}\\}$ such that

$\displaystyle||y_{i}-y_{j}||^{2}=m^{\prime}_{ij}=||x_{i}-x_{j}||^{2}-\beta$
(7)

for $0\leqslant i<j\leqslant d$.

Step 2: For $\delta\ll\beta$, we find $k$-dimensional vector
$c=(c_{1},\ldots,c_{k})\in\mathbb{R}^{k}$ and
$S_{2}=\\{z_{0},\ldots,z_{d}\\}\subseteq\operatorname{Spread}(c,[n])$ with
$||z_{i}-y_{i}||<\delta$ for $0\leqslant i\leqslant d$. Moreover,

$\displaystyle||z_{i}-z_{j}||^{2}=||x_{i}-x_{j}||^{2}-\beta\pm\varepsilon$

where $\varepsilon:=\varepsilon(\beta,d)\rightarrow 0$ as $\delta\rightarrow
0$.

Let $\eta:=\eta(\beta,d)>0$ be the positive real number given by Theorem 4.6,
let $\varepsilon<\min\\{\beta/64d^{2},\eta\\}$ and let
$\delta:=\delta(\eta,\rho)$ be sufficiently small. By Corollary 4.2, there
exist integers $n$, $k$, a $k$-dimensional vector
$c=(c_{1},\ldots,c_{k})\in\mathbb{R}^{k}$ with $||c||=\rho^{\prime}$ and a
simplex $S_{2}=\\{z_{0},\ldots,z_{d}\\}\subseteq\operatorname{Spread}(c,[n])$
such that $||y_{i}-z_{i}||<\delta$ for $0\leqslant i\leqslant d$. Thus, a
triangle inequality gives us that

$\displaystyle||y_{i}-y_{j}||-2\delta<||z_{i}-z_{j}||<||y_{i}-y_{j}||+2\delta.$
(8)

Hence, by combining (7) and (8)

$\displaystyle||x_{i}-x_{j}||^{2}-\beta+4\delta^{2}-4\delta||y_{i}-y_{j}||<||z_{i}-z_{j}||^{2}<||x_{i}-x_{j}||^{2}-\beta+4\delta^{2}+4\delta||y_{i}-y_{j}||.$

Since $||y_{i}-y_{j}||<2\rho^{\prime}$ and
$4\delta^{2}+8\delta\rho^{\prime}<\varepsilon$ for sufficiently small
$\delta$, we have that

$\displaystyle||x_{i}-x_{j}||^{2}-\beta-\varepsilon<||z_{i}-z_{j}||^{2}<||x_{i}-x_{j}||^{2}-\beta+\varepsilon$

Step 3: We find an almost regular simplex $S_{3}=\\{w_{0},\ldots,w_{d}\\}$
satisfying

$\displaystyle||w_{i}-w_{j}||^{2}=||x_{i}-x_{j}||^{2}-||z_{i}-z_{j}||^{2}=\beta\pm\varepsilon$

for all $0\leqslant i\neq j\leqslant d$. Furthermore, there exists a brick $B$
such that $S_{3}\subseteq B$.

This is an easy consequence of our preliminary results. By our choice of
$\varepsilon$, Corollary 4.5 guarantees that there exists simplex
$S_{3}=\\{w_{0},\ldots,w_{d}\\}$ such that

$\displaystyle||w_{i}-w_{j}||^{2}=||x_{i}-x_{j}||^{2}-||z_{i}-z_{j}||^{2}.$

Moreover, by Theorem 4.6, there exists a $\binom{d+1}{2}$-dimensional brick
$B$ such that $W\subseteq B$.

Step 4: We construct a simplex $F\cong S$ such that $F\subseteq
B\times\operatorname{Spread}(c,[n])$ and
$F\not\subseteq\operatorname{Spread}(c,[n])$ and apply Theorem 3.4

Let $F=\\{f_{0},\ldots,f_{d}\\}$ be the simplex defined by

$\displaystyle f_{i}=w_{i}*z_{i},$

where the symbol $*$ stands for the usual concatenation, i.e., if
$a=(a_{1},\ldots,a_{r})$ and $b=(b_{1},\ldots,b_{s})$, then
$a*b=(a_{1},\ldots,a_{r},b_{1},\ldots,b_{s})$. Hence,

$\displaystyle||f_{i}-f_{j}||^{2}=||w_{i}-w_{j}||^{2}+||z_{i}-z_{j}||^{2}=||x_{i}-x_{j}||^{2}$

for every $0\leqslant i,j\leqslant d$. Thus, the configuration $F$ is a copy
of $S$. Furthermore, by the construction of $F$ we have that

$\displaystyle F\subseteq W\times Z\subseteq
B\times\operatorname{Spread}(c,\mathbb{N}),$

where $B$ is a $\binom{d+1}{2}$-dimensional brick and
$\operatorname{Spread}(c,\mathbb{N})$ is a robust configuration (Claim 4.3).
Since $\rho(\operatorname{Spread}(c,\mathbb{N}))=\rho^{\prime}<\rho=\rho(F)$,
we obtain that $F\not\subseteq\operatorname{Spread}(c,\mathbb{N})$ and by
Theorem 3.4 the configuration $F\cong S$ is P-Ramsey. ∎

##  5\. Concluding remarks

The list of known Ramsey configurations is quite limited. Apart of simplices
and bricks, the most significant result is due to Kříž [K91] who proved that
regular polygons are Ramsey. Unfortunately, our method of robust
configurations does not apply here. This leaves us with the following
question:

######  Question 5.1.

Are regular polygons P-Ramsey?

Differently from Theorem 1.1, the proof in [K91] do not provide a density
result for regular polygons. Another interesting question would be to
determine if such result exists.

######  Question 5.2.

Let $F$ be a regular polygon. For every $\mu>0$, is there a configuration
$Y:=Y(F,\mu)$ such that any set $Z\subseteq Y$ of size $|Z|\geqslant\mu|Y|$
contains a copy of $F$?

Lastly, another direction of research would be to obtain sharp bounds for the
real number $\mu$ in the P-Ramsey definition. It is not hard to show that for
a configuration $X$ with $k$ points we cannot take $\mu>\frac{k-1}{k}$.
However, our proofs of Theorem 1.3 and 1.4 only gives $\mu=\frac{1}{4}$. It
would be interesting to close the gap for simplices.

######  Question 5.3.

Let $S$ be a $d$-dimensional simplex. What is the largest value of $\mu>0$
such that there exists a configuration $Y$ satisfying properties $(i)$ and
$(ii)$ of Definition 1.2?

## References