# Influence of the Reynolds number from $\bm{Re_{\tau}=150}$ to $\bm{210}$ on
size-dependent bipolar charging

S Jantač1 H Grosshans1 2 1 Physikalisch-Technische Bundesanstalt (PTB),
Braunschweig, Germany  2 Otto von Guericke University of Magdeburg, Institute
of Apparatus- and Environmental Technology, Magdeburg, Germany
<EMAIL_ADDRESS>
###### Abstract

We recently found wall-bounded turbulence to suppress and control bipolar
triboelectric charging of particles of identical material. This control is due
to fluid modifying the motion of light particles. Thus, the particles’ charge
distribution depends on their Stokes number distribution. More specifically,
fluid forces narrow the bandwidth of the charge distribution, and bipolar
charging reduces dramatically. Consequently, not the smallest but mid-sized
particles collect the most negative charge. However, the influence of the
Reynolds number or particle concentration on bipolar charging of polydisperse
particles is unknown. This paper presents the charging simulations of same-
material particles the in different wall-bounded flows. In a comprehensive
study, we vary the Reynolds number from $Re_{\tau}=$ $150$ to $210$ and the
particle number density from $4\times 10^{9}\ \mathrm{m}^{-3}$ to $1\times
10^{10}\ \mathrm{m}^{-3}$ to further explore the influence of the carrier flow
on bipolar charging. We model charge transfer based on the balance of
transferable charge species. Such species can represent adsorbed ions
transferred during collisions or free electrons captured into a lower energy
state on the other surface. The turbulent flow is modeled via Direct Numerical
Simulations (DNS) and is coupled to the particulate phase modeled via the
Discrete Element Method (DEM). Overall, our multiphysics approach couples the
fluid dynamics, electric field, triboelectric charging, and particle momentum
into one complex simulation.

##  1 Introduction

Triboelectric charging causes a build-up of charge in particulate systems. In
industrial environments, the excessive build-up causes unwanted agglomeration
and wall-sheeting in the fluidized beds [1, 2, 3, 4, 5], pneumatic conveyors
[6, 7, 8, 9, 10], or during powder handling in general [11]. This is an
especially severe problem in the pharmaceutical industry [12, 13, 14, 15]
because the formulation of powders is strictly given and often cannot be
changed to reduce the charging. Triboelectric charging also plays a vital role
in many natural phenomena. Like volcanic eruptions, where the charging is one
of the contributors to the formation of lighting, and the charge build-up
hinders the correct prediction of tephra behavior [16, 17]. It was shown that
during planetary formation [18], the electrostatic forces play an essential
role, and thus accurate estimation of the charge build-up is vital.

In all the above-mentioned processes, the charge is transported due to a
combination of inter-particle and particle-wall collisions. The particle-wall
collisions typically occur between objects of different materials, a situation
that can be described by models that use a difference in the material property
as a driving force for the charge transfer [19]. However, inter-particle
collisions often occur between particles of identical material; thus, those
models cannot describe experimentally observed bipolar charging in
polydispersed systems of identical material [20, 21]. New mosaic-based models
were proposed that can explain those experiments based on the statistical
variation of properties on otherwise identical materials [22, 23]. However,
they describe the charging only at the nano/mesoscale.

Recently, we expanded our research to encompass turbulent flows with the
bipolar charging of identical materials and unveiled how turbulence
fundamentally alters the charge distribution [24]. Our investigation revealed
a major shift in highly polydisperse systems, specifically when the smallest
particles possess a Stokes number of $\approx 1.5$ or less. Surprisingly, in
such cases, the mid-sized particles carry the most negative charge, while the
smallest particles exhibit a negligible charge, a departure from the expected
behavior. Consequently, this significantly narrows down the charge
distribution in the polydisperse systems. Our analysis indicates that
turbophoresis is responsible for this reduction, arising from the decreased
local polydispersity within the system. Although our initial study was limited
to specific flow conditions, we have now demonstrated that these earlier
findings extend to other flow scenarios as well.

##  2 Simulation setup

Our simulation couples the DNS with the DEM approach to accurately simulate
the particle trajectories. The fluid phase is fully resolved DNS simulation
which is coupled with the particle phase [25, 26] . The particle phase is
simulated via DEM, and we consider the following forces: The electrostatic
force between all particles and grounded walls of the channel, the drag force,
and the lift force in the turbulent channel.

The domain was set up as a channel flow, i.e., boundaries of the system in the
$y$-axis were used as walls (zero slip conditions), the boundaries in the
$z$-axis were set to be periodic, and boundaries in the $x$-axis were set as
periodic. The dimension of the domain was $6H\times H\times 2H$ where $H=4$ cm
[see Fig. 1]. The corresponding grid resolution was $256\times 144\times 144$,
and the size of the cells in the $x$ and $z$ axis is uniform. This domain
setup mimics a section of the pneumatic conveyors.

Figure 1: DNS simulation of a channel flow. On the left is the side view where
the airflow is traveling from left to right. On the right is the front view of
the simulation domain.

Our charging model is based on the balance of charge carriers initially
homogeneously distributed on the surface. Lacks formulated this model type
[22], and Grosjean formulated a newer discretized version [27, 28]. However,
we will use a generalized version of those models formulated by Konopka and
Kosek [29].

Figure 2: Typical bipolar charging of the polydisperse system in the absence
of ambient flow. The color represents that particle size. a) The largest
particles are most positively charged, and the smallest are most negatively
charged. The net charge in the simulation is conserved and thus is zero
through the simulation (see the dashed line). b) The charging rate depends on
the charge carriers’ concentration; the charging ceases when the charge
carriers are depleted

.

We assume that the surface consists of two types, one is the charge carrier,
and the second one is the free spot where the charge carrier can be
spontaneously transferred (relaxed). The surface resistivity prevents any
movement of the charge carrier. Thus, the charge carrier can be transported
and subsequently relaxed only when it collides with the free spot on the other
surface. Once the charge carrier is transported, we assume it is stable and
does not participate in subsequent collisions. Additionally, we assume that
the concentration of transferable charge carriers is much lower than that of
free spots. This means that we can formulate a balance equation of only charge
carriers to describe the net charge transfer during collision:

$\Delta q_{i}=ze(c_{\mathrm{s},j}-c_{\mathrm{s},i})A_{\mathrm{c,max}}^{i,j},$
(1)

where $z$ is the charge number of the charge species, $e$ is the elementary
charge, $c_{\mathrm{s},j}$ is the concentration of charge species on $j$-th
particle, and finally $A_{\mathrm{c,max}}^{i,j}$ is the maximal collision
contact area of $i$-th and $j$-th particle, which can be calculated as follows
[30]:

$A_{\mathrm{c,max}}^{i,j}=\pi\left(\frac{5m_{i,j}r_{p,i,j}^{2}}{4E_{i,j}}\right)^{2/5}v_{n}^{4/5},$
(2)

where $m_{i,j}$ is the effective mass, $r_{p,i,j}$ is the effective radius,
and $E_{i,j}$ is the effective elastic modulus of colliding particles, which
are calculated as the reciprocal value of the harmonic sum of $i$-th and
$j$-th particles properties. By balancing the charge carriers on the particle
level, we additionally assume that enough time has passed for the particle to
be randomly reoriented for the subsequent collision; this allows us to
describe the charge transfer only as a function of charge carrier
concentration.

Summing the charge transfer over $n$ collisions yields a net charge on each
particle:

$q_{i}=q_{i,t_{0}}+\sum_{i=1}^{n}\Delta q_{i}(n),$ (3)

where $q_{i,t_{0}}$ is the initial charge of particle. To illustrate the
charging of the polydisperse system with the above-mentioned model, we show
the result of simulations in the absence of flow (Figure 2). The absence of
flow simplifies the description and allows validation of our simulations with
previously published works [29, 31, 32]. We assume that particles are
initially uncharged and have uniform concentrations of charge carriers. This
causes the initial few collisions of particles to transfer little charge
because an equal amount of charge carriers are transported between colliding
particles. However, as charge carriers start to deplete, the rate of charging
increases, and since small particles deplete the charge carriers faster than
large ones [see Figure 2], they start to charge negatively. In contrast, large
particles charge positively because they are the donors of the negatively
charged charge carriers. Electrons or adsorbed ions are examples of negatively
charged carriers.

##  3 Results

Before we enabled the charging of the particles, we let both the fluid field
and particle concentration fully converge. After convergence, we initialized
the triboelectric model. The initial values of particles were set as follows:
zero initial charge ($q_{i,t_{0}}=0$ C ), uniform charge carrier density
($c_{\mathrm{s},t_{0}}=10\ \mathrm{\upmu m}^{-2}$), the charge number of the
carrier was assumed to be equivalent to electron ($z=-1$), the elastic modulus
of particles was set to $E=1\times 10^{8}\ \mathrm{Pa}$.

The charge build-up is shown in Fig. 2. The largest particles tend to charge
positively. And small particles tend to charge negatively. The decay of charge
carriers has an exponential trend, as shown in Fig. 2. However, as shown in
our recent paper [24], the charge distribution is qualitatively changed when
the coupling between the fluid and particles is considered. Here we study the
effect of friction Reynolds number and particle number density. In the flowing
studies, we evaluate the charge distribution when the average concentration of
charge species decreases by a characteristic value of $1/\mathrm{e}$ of
initial concentration (here, the $\mathrm{e}$ is the Euler number).

###  3.1 Friction Reynolds number

Figure 3: a) The bipolar charging is significantly suppressed when $Re_{\tau}$
is varied from 150 to 210. The full lines denote the results when
$Re_{\tau}=210$, and the dashed lines when $Re_{\tau}=150$. For clarity, we
show every second case of $\sigma_{St}$, also note that the y-axis is scaled
in the units of the elementary charge. b) the standard deviation of charge as
a function of particle size distribution measured as a $\sigma_{St}$ shows an
increase in the width of the charge distribution with an increase in
$Re_{\tau}$. Values of $Re_{\tau}=180$ are taken from our recent paper [24].

In this section, we studied the effect of the friction Reynolds number on
bipolar charging. We set the particle number density to be $4\times 10^{9}\
\mathrm{m}^{-3}$, and varied $Re_{\tau}$ from 150 to 210. Additionally, we
varied the particle size distribution from monodisperse to polydisperse, where
the smallest particles were 10 µm in diameter and the largest 140 µm in
diameter. The results of those simulations show that in the studied range of
$Re_{\tau}$, the qualitative shape of the charge distribution remains the same
[see, Fig. 3]. However slight increase in the width of the charge distribution
was observed. To quantify the width of the charge distribution, we evaluated
the standard deviation of the charge on particles $\sigma_{q_{p}}$ and a
standard deviation of the Stokes number of particles in the system
$\sigma_{St}$. The relative difference between the standard deviation of
particle charge in all particle size distributions was around 7 % [see Fig.
3].In fact, the values of $\sigma_{q_{p}}$ appear to be proportionally shifted
with increasing $Re_{\tau}$. Such a consistent shift indicates that the
distribution of inter-particle collisions is very similar in both cases, only
the frequency of collision increases.

We confirmed that the bipolar charging of identical materials is suppressed in
the studied range of $Re_{\tau}$ by measuring the width of the charge
distribution at the evaluation point
$c_{\mathrm{s}}=c_{\mathrm{s},t_{0}}/\mathrm{e}$, but the time required to
reach this point was reduced from $\approx 3.6$ s when $Re_{\tau}=150$ to
$\approx 3.0$ s when $Re_{\tau}=210$.

###  3.2 Particle number density

The particle number density plays a stronger role in bipolar charging.
Studying two cases of the particle number density of $5\times 10^{-9}\
\mathrm{m^{-3}}$, and $10\times 10^{-9}\ \mathrm{m^{-3}}$, showed that the
charge distribution is significantly changed [see Fig. 4]. Neutral particles
are smaller when the number density is larger. The charge of most negative
particles is higher for higher number density. This indicates that inter-
particle collisions are significantly changed [see Fig. 4]. In the denser
case, the particles of $St_{d_{p}}=0.4$ and $St_{d_{p}}=0.7$ collided more
frequently with all particles; this causes them to lose charge carriers faster
than other particles, making them negatively charged. The average collision
ratio is similar for both cases for particles of $St_{d_{p}}=1.6$ which makes
them similarly charged [see the intersection of green lines at Fig. 4]. A
particle with $St_{d_{p}}\geq 2.1$ collided more frequently with smaller
particles and less frequently with larger particles. Since in later stages of
charging, the smaller particles have a lower concentration of charge carriers
than large particles [see Fig. 2], the large particles in denser cases charged
more positively. This indicates that bipolar charging will be more significant
in systems with high particle number density.

Figure 4: a) The charge distribution for two most polydisperse cases, when
$Re_{\tau}=210$. The increase in volumetric density causes increased charge
accumulation. b) The distribution of inter-particle collisions for
$\sigma_{St}=1.67$ shows that in denser cases, the smaller particles collide
more frequently with particles of all sizes, and large particles collide less
frequently with particles of similar size.

##  4 Conclusions

Triboelectric charging causes severe problems in industry and frequently
occurs in nature. Current triboelectric models are based on the
nano/mesoscale, which limits their prediction to, at best, two-particle
charging and naturally cannot describe emergent properties. However, the
above-mentioned phenomena are macroscopic and therefore include a vast number
of particles whose collective charging behavior cannot be described simply by
fundamental models. To improve our understanding of such a system, we simulate
a large-scale powder system in a change flow, where particles of identical
material properties charge bipolarly based on recent mosaic models. Our
simulation couples DNS and DEM in a four-way coupling which includes the
coupling of the gas and particle phase, the particle position is coupled with
the electrostatic forces created by the accumulated charge, and lastly, the
particle charge is dependent on the particle collisions and positions.

Our research has revealed that turbulence is crucial in suppressing bipolar
charging across a broader spectrum of flow conditions and particle number
densities. Specifically, the width of the charge distribution is influenced by
the friction Reynolds number, but it doesn’t significantly alter the
previously reported qualitative change in the charge distribution as described
in [24]. As the Reynolds number increases, collision frequency rises
proportionally, leading to a shift in particle charge standard deviation.
However, the particle number density has a more pronounced impact on bipolar
charging. With higher particle number densities, the charge distribution
undergoes more substantial changes, causing smaller particles to charge
negatively and larger particles to charge positively due to differences in
collision frequencies. Overall, our findings demonstrate that in turbulent
environments, bipolar charging experiences suppression at least within the
range of $Re_{\tau}$ 150 to 210 and at particle number densities between
$5\times 10^{-9}\ \mathrm{m^{-3}}$ and $10\times 10^{-9}\ \mathrm{m^{-3}}$.

The findings suggest that understanding and managing triboelectric charging
are crucial for optimizing particulate systems in industrial processes. Proper
consideration of charging effects can lead to better control and design of
such systems. Our conclusions can be extended to polydisperse turbulent flows
inside volcanic plumes or dust storms.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 947606 PowFEct).

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# Deep two-phase, hemispherical magma oceans on lava planets

Charles-Édouard Boukaré Institut de Physique du Globe de Paris, 1, rue
Jussieu, Paris CEDEX 05, 75238, France  Nicolas B. Cowan Department of Earth
and Planetary Sciences, McGill University, 3450, rue University, Montréal,
Québec, H3A 0E8, Canada  Department of Physics, McGill University, 3600, rue
University, Montréal, Québec, H3A 2T8, Canada  James Badro Institut de
Physique du Globe de Paris, 1, rue Jussieu, Paris CEDEX 05, 75238, France

###### Abstract

Astronomers have discovered a handful of exoplanets with rocky bulk
compositions but orbiting so close to their host star that the surface of the
planet must be at least partially molten. It is expected that the dayside of
such “lava planets” harbors a rock vapor atmosphere that flows quickly towards
the airless nightside—this partial atmosphere is critical to the
interpretation of lava planet observations, but transports negligible heat
towards the nightside. As a result, the surface temperature of the magma ocean
may range from 3000 K near the sub-stellar point down to 1500 K near the
day–night terminator. We use simple models incorporating the thermodynamics
and geochemistry of partial melt to predict the physical and chemical
properties of the magma ocean as a function of the distance from the sub-
stellar point. Our two principal findings are that 1) the dayside magma ocean
is much deeper than previously thought, probably extending down to the
core–mantle boundary in some locations, and 2) much of the dayside is only
partially molten, leading to gradients in the surface chemistry of the magma
ocean. These findings have important implications for the dynamics of the
magma ocean as well as the composition and dynamics of the atmosphere.

Exoplanets - Magma Ocean - Lava Worlds

##  1 Introduction

Astronomers have recently discovered a new class of planet, planets with bulk
densities suggestive of terrestrial composition, but orbiting so close to
their host star that the dayside equilibrium temperature exceeds the solidus
temperature of silicates, and hence must be molten. Notable examples include
CoRoT-7b (Léger et al., 2009), Kepler-10b (Batalha et al., 2011), 55 Cnc e
(Demory et al., 2011; Winn et al., 2011), and K2-141b (Barragán et al., 2018;
Malavolta et al., 2018). These “lava planets” are predicted to have a
permanent magma ocean with an overlying rock-vapor atmosphere (Schaefer &
Fegley, 2009). As with most short-period exoplanets, lava planets are expected
to be tidally-locked into synchronous rotation (Léger et al., 2011). For a
review of lava planets, read Chao et al. (2021).

###  1.1 Atmospheres of Lava Planets

In the likely event that a lava planet has already lost its volatiles to
space, then it cannot maintain a steady-state atmosphere on its nightside
(Léger et al., 2011). Instead, it will have a partial atmosphere of rock vapor
on the dayside that blows towards the airless nightside, cools and condenses
onto the surface, only to be returned to the sub-stellar point via magma ocean
currents and solid-state flow near the shores of the ocean (Castan & Menou,
2011; Kite et al., 2016; Nguyen et al., 2020). If a lava planet somehow
retains volatiles over gigayears, then it may have a global atmosphere and
relatively homogenized surface temperatures (Hammond & Pierrehumbert, 2017).
In the current paper we adopt the majority view, namely that lava planets have
a day-to-night temperature contrast of more than 2000 K, with a dayside magma
ocean and an airless nightside.

Fortunately, the high temperatures of lava planets make them amenable to
observational studies. Transits, eclipses and phase variations of lava planets
have been observed with the CoRoT, Kepler, and Spitzer space telescopes (Léger
et al., 2009; Batalha et al., 2011; Demory et al., 2016), and further
observations have been approved for the Hubble and James Webb space telescopes
(e.g., Dang et al., 2021; Espinoza et al., 2021; Hu et al., 2021; Quinn et
al., 2021). These measurements constrain the surface temperature of a lava
planet as a function of longitude, the atmospheric composition, and even the
atmospheric temperature structure (Zilinskas et al., 2020, 2021; Nguyen et
al., 2020). While remote sensing can only directly probe an exoplanet’s
atmosphere and surface, one may sometimes indirectly infer properties of its
interior (Cowan, 2014).

The atmospheric composition and dynamics on a lava planet depend on the
composition and dynamics of its magma ocean. To first approximation, the
atmosphere at some location on a lava planet is dictated by the saturation
vapor pressure above the liquid surface (Castan & Menou, 2011). The spatial
variations in temperature—and hence vapor pressure—is the dominant source of
atmospheric dynamics in its atmosphere.

With observations of their atmospheres becoming available, it is therefore
timely to produce model predictions for the magma oceans on lava planets.

###  1.2 Interiors of Lava Planets

The magma ocean on a lava planet is governed by the same geochemistry and
geophysics as the magma oceans thought to have existed at the formation of
rocky worlds in the Solar System (Elkins-Tanton, 2012), but with drastically
different boundary conditions. First of all, the magma ocean on a lava planet
is in a steady state: it is not gradually cooling down. Secondly, a lava
planet’s ocean is heated from above. Lastly, the magma ocean on a lava planet
is horizontally heterogeneous, going from fully molten near the sub-stellar
point ($T\approx 3000$) to completely solid near the day–night terminator
($T\approx 1000$)—in other words, lava planets cannot be treated as spherical
cows.

Previous models of lava planet interiors assumed that the dayside magma ocean
is shallow (Léger et al., 2011; Kite et al., 2016; Nguyen et al., 2020). Using
a liquidus temperature gradient of about 30 K GPa-1 (Solomatov, 2000),
previous authors predicted that the temperature of the magma pond intersects
the planet’s liquidus temperature at a depth of about 70 km (e.g., see
appendix B of Kite et al., 2016).

However, for a multi-component system such as the bulk silicate Earth (BSE),
the liquidus does not characterize the temperature at which magma becomes
solid. The magmas composed of several oxides such as MgO, FeO, SiO2 or CaO,
fully solidify only when the temperature drops below the solidus. For
temperatures between the solidus and the liquidus, the magma is partially
molten. There is a rheological transition at about 50% of melt fraction where
magma starts to behave as a solid rather than a liquid (e.g., Abe, 1993;
Lejeune & Richet, 1995; Costa, 2005). We show here that even though the
temperature in the magma pond intersects the liquidus at shallow depth, the
temperature in the deeper part of the planet remains close the liquidus due to
latent heat effects.

Major advances in laser-heated diamond anvil cell (LHDAC) now gives direct
access to pressures and temperatures relevant to the deep interior of
terrestrial planets. Recent measurements of liquidus and solidus temperature
in LHDAC (e.g., Fiquet et al., 2010; Andrault et al., 2011, 2012, 2014; Tateno
et al., 2014, 2018; Pradhan et al., 2015) show that the temperature difference
between liquidus and solidus for Earth-like material can be as small as 400
K(Andrault et al., 2011) and as large 1000 K (Fiquet et al., 2010). Using
these recent experimental data at high pressures, we show that hemispherical
magma oceans in lava planets are most likely deep and probably extend down to
the core-mantle boundary for surface temperatures as low as 1900 K.

In section 2, we present our assumptions and describe how we compute the
interior temperature profile of a lava planet. In section 3 we present our
results regarding the magma ocean depth, the location of the shore, and
potential chemical gradients within the magma ocean due to partial
crystallization. In section 4, we speculate about the global internal dynamics
of lava planets, its effect on atmospheric composition and future
spectroscopic observations.

##  2 Model

We consider a tidally locked planet of about 1.5 Earth radii that orbits close
to its parent star. Since the overlying atmosphere is optically thin, the
planet’s surface temperature is well approximated by local radiative
equilibrium (Castan & Menou, 2011; Léger et al., 2009; Nguyen et al., 2020).
The large angular size of the star as seen from the planet leads to an
extended penumbra region and less than a hemisphere in the dark (Léger et al.,
2011; Nguyen et al., 2020) and we adopt a temperature at the sub-stellar point
appropriate for K2-141b, $T_{\rm ss}=3000$ K. As we shall see, our model can
easily apply to cooler lava planets. We also assume that the composition of
the lava planet is that of the Bulk Silicate Earth (BSE; Palme & O’Neill,
2003).

On the day side, lava planets are expected to be fully molten (e.g., Kite et
al., 2016) for surface temperature greater than 1700 K, forming a magma ocean.
As temperature decreases toward the night side, the planetary surface is
expected to transition from liquid to solid. The location of the shore is
determined by (1) the variation of the surface temperature with respect to the
angular distance from the sub-stellar point, and (2) the melting temperature
of silicates at low pressure. Similarly, the depth of the magma ocean is
constrained by (1) the interior temperature profile, and (2) the melting
temperature of silicates at high pressures.

The interior thermal structure of the planet has a first order control on the
planet’s dynamics as it governs the relative size of the magma ocean compared
to the solid mantle. Both the depth and lateral extent of the magma ocean have
a primary role in the atmosphere-interior coupling. Indeed, the rates of heat
and mass transfer differ by several orders of magnitude between solid and
liquid states. For the sake of simplicity, we assume that the planet’s
interior temperature follows an adiabat. This choice is motivated by the fact
that the planet interior is expected to convect vigorosly. Convective motions
are driven by temperature differences between the surface and the core-mantle
boundary (CMB) as well as temperature contrast between the day-side and the
night side. For comparison, the modern Earth has a surface temperature of
about 300 K and CMB temperatures range between 4000 and 5000 K. For a planet
with higher surface temperature such as lava planets, it is unlikely that the
CMB temperature is lower than the Earth’s at the same age. Hence the CMB
temperature on a lava planet is expected to greatly exceed the surface
temperature, even at the substellar point, certainly generating a flow of heat
from the interior to the surface and possibly driving convective motions. In
this study, we aim to estimate a pseudo steady-state interior thermal
structure if heat is transported by convection. We neglect multiphase
mechanical processes such as crystal settling, solidification dynamics, and
melt percolation on the planet’s structure (e.g., Maas & Hansen, 2015; Boukaré
& Ricard, 2017).

###  2.1 Liquidus and solidus temperature

Two temperatures are required to describe the melting behavior of silicate
materials. Above the liquidus temperature, silicates are fully liquid. Below
the solidus temperature, silicates are fully solid. Silicates are partially
molten for temperatures that lie between the solidus and the liquidus. This is
different from “pure species” such as SiO2 or MgO whose phase change occurs at
a single temperature. Both liquidus and solidus temperature are functions of
pressure and composition.

High pressure liquidus and solidus temperatures of the BSE are an area of
active research (Fiquet et al., 2010; Andrault et al., 2011, 2012, 2014). To
propose a conservative estimate of the hemispherical magma ocean depth, we use
the experimental data of Fiquet et al. (2010)—the highest reported liquidus
temperature. Our model thus provides a lower bound on the magma ocean depth.
The experimental data of Andrault et al. (2011), for example, suggest a lower
liquidus temperature and hence favor a deeper magma ocean.

The solidus and liquidus temperatures are fitted to the experimental data of
Fiquet et al. (2010) and Zhang & Herzberg (1994). The liquidus, $T_{l}$, and
solidus, $T_{s}$, are plotted in Figure 1 and given by

$T_{l}(P)=2000~{}{\rm K}\left(0.1169\left(\frac{P}{\rm
GPa}\right)+1\right)^{0.32726}$ (1) $T_{s}(P)=1674~{}{\rm
K}\left(0.0971\left(\frac{P}{\rm GPa}\right)+1\right)^{0.351755}.$ (2)

Figure 1: High-pressure liquidus (red) and solidus (blue) temperature for
Earth-like mantle compositions. Above the liquidus temperature, silicates are
liquid. When the temperature lies between the liquidus and the solidus,
silicates are partially molten. Silicates fully solidify when the temperature
drops below the solidus. Colored symbols show recent experimental measurements
(Fiquet et al., 2010; Andrault et al., 2011, 2014; Zhang & Herzberg, 1994) and
theoretical calculations (Boukaré et al., 2015; Usui & Tsuchiya, 2010) of the
liquidus and solidus temperatures at high pressures. Discrepancies among data
in the literature arise from differences in approaches such as silicate
compositions or melt detection techniques. For the sake of simplicity, we
approximate the liquidus and solidus curves for Earth-like composition by the
red and blue lines, Equations (1) and (2). On the left, pressure-depth
conversion is shown for the Earth and a $1.5R_{\oplus}$ super-Earth. Note that
the melting curve of pure SiO2, shown by the green asterisks, substantially
overestimates the liquidus temperature of the Bulk Silicate Earth.

###  2.2 Isentropic temperature profiles

The temperature profile in the fully molten region is calculated using the
isentropic (and thus adiabatic) temperature gradient of a one phase system,

$\left(\frac{\partial T}{\partial P}\right)_{S}^{\rm op}=\frac{\alpha
V}{C_{p}}T$ (3)

where $\alpha$ is the thermal expansion, $V$ the volume, and $C_{p}$ the
thermal capacity of the silicate melt. A self-consistent integration of Eq (3)
requires a thermal equation of state (EoS) for silicate melt. We use

$\displaystyle P(V,T)=$ $\displaystyle
3K_{0}f^{-2}(1-f)e^{\frac{3}{2}(K^{\prime}-1)(1-f)}$ (4)
$\displaystyle+\alpha_{0}K_{0}f^{3q-3}(T-T_{0}),$

where the dimensionless length is $f=(V/V_{0})^{1/3}$, $K_{0}$ and
$\alpha_{0}$ are the isothermal bulk modulus and thermal expansion at the
reference conditions ($V_{0}$ and $T_{0}$), $K^{\prime}=(\partial
K_{0}/\partial P)_{V}$ and $q$ are constants. The first term on the right-hand
side of Eq (4) corresponds to the Vinet-Rydeberg isothermal EoS (Vinet et al.,
1987), while the second term is the thermal pressure as initially proposed for
solids (Anderson, 1979). Whereas the description of thermal pressure for
solids requires positive values of $q$, liquids are generally described by
negative values of $q$ (Asimow & Ahrens, 2010; Boukaré et al., 2015). We
fitted thermodynamic data of Boukaré et al. (2015) with Eq 4. We use $V_{0}=$
26.27 cm3 mol-1, $T_{0}=$ 298 K, $\alpha_{0}=10.8\times 10^{-5}$ K-1, $K_{0}=$
17.58 GPa, $K^{\prime}=$ 6.9 and $q=$ -3.37.

In the partially molten region, the temperature profile follows a two-phase
adiabat (e.g., Miller et al., 1991; Asimow & Ghiorso, 1998) akin to moist
adiabats in planetary atmospheres,

$\left(\frac{\partial T}{\partial P}\right)_{S}^{\rm
tp}=\frac{\left(\frac{\partial T}{\partial P}\right)_{S}^{\rm
hom}-\frac{T\Delta S}{C_{p}}\left(\frac{\partial F}{\partial
T}\right)_{P}\left(\frac{\partial T_{m}}{\partial
P}\right)_{F}}{1-\frac{T\Delta S}{C_{p}}\left(\frac{\partial F}{\partial
T}\right)_{P}},$ (5)

where $\Delta S$ is the entropy difference between the liquid and the solid
phase, $F$ is the melt fraction, $(\partial F/\partial T)_{P}$ is the
variation of the melt fraction with temperature at constant pressure, and
$(\partial T_{m}/\partial P)_{F}$ is the variation of the equilibrium
temperature with pressure at constant melt fraction. Rigorous application of
Eq (5) requires detailed self-consistent phase diagrams to constrain the
latter parameters. Latent heat goes as $T\Delta S\propto TR$ and thermal
capacity as $C_{p}\propto 3R$, so we assume that $T\Delta S/C_{p}\propto T$,
following Miller et al. (1991). To validate this assumption, we ran
calculations using the thermodynamic model of Boukaré et al. (2015),
confirming that $\Delta S/C_{p}$ is of order unity. We use the crude
assumption that melt fraction varies linearly between the liquidus and the
solidus at fixed pressure, so that

$\left(\frac{\partial F}{\partial T}\right)_{P}=\frac{1}{T_{l}(P)-T_{s}(P)},$
(6)

where $T_{l}(P)$ and $T_{s}(P)$ are liquidus and solidus temperature at
pressure $P$, respectively.

##  3 Results

###  3.1 Magma ocean depth

Figure 2: Interior temperature of lava planets for four different surface
temperatures: 1700 K, 1900 K, 2400 K, and 3000 K. The temperature increase
with depth is due to the isentropic compression of the material. In regions
hotter than the liquidus, the temperature follows a one-phase, homogeneous,
isentrope (or adiabat). When the temperature is between the liquidus and the
solidus, entropy conservation must account for the latent heat of fusion. In
this region, the temperature follows a two-phase isentrope, akin to the moist
adiabat of planetary atmospheres. For a surface temperature of 3000 K, the
silicate mantle is expected to be fully molten for a planet the size of Earth
or a $1.5R_{\oplus}$ super Earth (see also Solomatov, 2007; Lebrun et al.,
2013). For a surface temperature of 2400 K, the geotherm intersects the
liquidus at a depth of about 100 km for a $1.5R_{\oplus}$ super Earth.
However, the geotherms remains very close to the liquidus at greater depth
such that the interior is still expected to be mostly molten. The lava planet
interior behaves as a liquid, i.e. has a melt fraction greater than 50%, for
surface temperature above 1700 K (see Figure 3).

Our calculations show that the magma ocean on the permanent dayside of a lava
planet is deeper than previously thought (see Figure 2). For a surface
temperature of 3000 K, the interior temperature remains above the liquidus
temperature at least up 130 GPa, i.e, 1200 km in a $1.5R_{\oplus}$ super-
Earth. For a surface temperature of 2400 K, the interior temperature
intersects the liquidus at about 15 GPa, i.e, 150 km in a $1.5R_{\oplus}$
super-Earth, but it remains close to the liquidus at higher pressures. This is
due to latent heat effects that bend adiabats such that they remain
approximately parallel to the liquidus. The latter is controlled by the second
term on the numerator of the right-hand side of Eq (5). For a surface
temperature of 1700 K, the interior temperature is halfway in between the
liquidus and the solidus temperature from the surface down to 120 GPa. These
results are in excellent agreement with previous Earth’s magma ocean model
(e.g., Abe, 1997; Solomatov, 2000; Thomas et al., 2012; Lebrun et al., 2013;
Monteux et al., 2016).

Figure 3: Interior melt fraction as a function of depth for four surface
temperatures: 3000 K, 2400 K, 1900 K, and 1700 K. Rheological transition
between solid-like and liquid-like behavior occurs at about 50% melt fraction.
The bulk of the magma ocean therefore behaves as a sluggish fluid—not a
solid—for surface temperature as low as 1700K.

We use Eq (6) to determine the melt fraction as a function of depth for
different surface temperature, shown in Figure 3. Our calculations indicate
that the melt fraction in the magma ocean stays above 50% for surface
temperatures hotter than 1700 K. As the rheologicical transition between solid
and liquid occurs at about 50% of melt fraction, these results show that the
hemispherical magma ocean behaves as a liquid for surface temperature as low
as 1700 K. For melt fraction close to the rheologicial transition, the
multiphase mixture is expected to flow as a sluggish fluid—not as a solid.

Our temperature and melt fraction profiles (Figures 2 and 3) can be used as a
first-order estimation of the interior structure of lava planets in two cases.
Our findings apply either to (1) multiple lava planets with different surface
temperature at the sub-stellar point, or (2) to different locations in the
same lava planet. If we neglect thermal mixing in the deep interior, these
profiles can be used as a crude proxy for evaluating the effects on the
interior thermal structure of lateral temperature variations imposed at the
surface. Adopting this strong assumption for K2-141b, the interior profiles
for surface temperatures of 1700 K, 2000 K, 2400 K, and 3000 K correspond to
angular distances from the sub-stellar point of $\theta=90$°, $\theta=80$°,
$\theta=60$°, and $\theta=0$°.

###  3.2 Where is the magma ocean shore ?

Figure 4: Lateral variation of the surface composition of tidally-locked lava
planets. Concentration of SiO2 (yellow), FeO (red), Na2O (green), H20 (blue),
and K2O (black) as a function of surface temperature (bottom axis) or distance
from the sub-stellar point on K2-141b (top axis). There is no well-defined
magma ocean shore but rather a gradual transition from the molten to the solid
hemisphere. In this partially molten region, the composition of the magma
evolves as precipitating minerals do not have the same composition as the
melt. Depending on the style of convective dynamics, these chemical
heterogeneities can be mixed back into the convective region or permanently
from the surface. For instance, enrichment in FeO would promote the formation
of very dense melt that might sink to the bottom of the magma ocean (Kite et
al., 2016). Additionally, the concentration of volatile species such as H2O
and CO2 can eventually reach the saturation limit in the melt near the
solidus. When the melt is saturated, any increase in concentration results in
the formation of bubbles that degas at the surface of the planet, with
possible implications for transit observations of lava planets.

Temperature decreases away from the sub-stellar point. Eventually, the
temperature drops below the liquidus. We use the software MELTS 1.2.0 to
compute the crystallization sequence of magma with a BSE composition at
shallow depth, i.e., 250 bar. For the sake of simplicity, we only consider the
case of batch crystallization where crystals remain in chemical equilibrium
with the melt during solidification. These thermodynamic calculations allow us
to predict more precisely the evolution of melt composition in major and minor
elements as well as liquidus and solidus temperature at the surface of the
planet. These temperatures constrain the location of the shore.

For Kepler K2-141b, with a sub-stellar surface temperature of 3000 K, the
transition from fully liquid to partially molten occurs near
$\theta=80^{\circ}$. The solidus temperature is reached at
$\theta=100^{\circ}$. There is no well-defined shore at the surface of lava
planets but rather a gradual transition from the molten to the solid
hemisphere. It must be noted that the partially molten region will be larger
for lava planets with more modest sub-stellar temperatures. For example,
K2-22b has a sub-stellar surface temperature of 2000 K resulting in a
partially molten region that extends from $\theta=0^{\circ}$ to
$\theta=80^{\circ}$.

###  3.3 Chemical fractionation in the magma ocean

Our calculations of the magma crystallization sequence (Figure 4) offers the
possibility to predict compositional variation induced by solidification at
the surface of a lava planet. Upon solidification, incompatible species stay
in the melt instead of precipitating as minerals. Incompatible species such as
Na2O, K2O or H2O are thus concentrated in the melt as crystallization proceeds
(see Figure 4). Since melt fraction is primarily controlled by temperature, we
propose that lateral variation of surface temperature generates compositional
heterogeneities in lava planets by incongruent melt solidification.

Moreover, our modeling indicates that an additional fluid phase composed
mostly of H2O exsolves from the melt as the temperature approaches the
solidus. Here, H2O will degas because its concentration in the melt has
increased to the point of saturation. This can be seen in Figure 4 in the
plateau of the magma water content close to the solidus temperature. This
process is different from atmosphere formation by liquid-vapor equilibrium
where pressure is set by vapor exsolution itself. In the planet interior,
pressure is set by the geostatic pressure. In that case, H2O degassing is more
akin to the behavior of salt in water. When water is saturated in salt, any
increase in salt content result in salt exsolution.

##  4 Discussion

###  4.1 Implications for lava planet interior structure and dynamics

Figure 5: Schematic of the interior structure and dynamics of a lava planet.
The hemispherical magma ocean on the dayside could extend down to the
core–mantle boundary. The night-side mantle is solid. A partially molten
region probably connects the two hemispheres. The illustrated pseudo steady-
state structure involves fluid motions. The solid-liquid interfaces are not
static, rather rocky materials are expected to flow through the solid-liquid
interface, melting or solidifying at these boundaries depending on the flow
direction. The surface temperature gradient drives magma currents between the
day and night sides. Temperature differences between core–mantle boundary and
the planetary surface are expected to drive convection of the rocky mantle.
Geodynamic models are required to explore the horizontal convective dynamics
as well as the efficiency of solid-liquid phase separation and chemical
mixing.

Based on these thermodynamics calculations, we propose a new model for the
interior structure of lava planets (figure 5). On the day side, the magma
ocean is at least 1000 km deep, or 135 GPa, and probably extends down to the
core-mantle boundary. Future high-pressure studies are required to constrain
liquidus temperatures at conditions relevant for the deep mantles of Super-
Earth, up to about 400 GPa for a $1.5R_{\oplus}$ super-Earth with core-mantle
ratio similar to the Earth. Crystal fraction increases progressively towards
the night side, enriching the melt in incompatible elements. This process may
be able to progressively differentiate the magma ocean depending on the
efficiency of phase separation and chemical mixing in the interior.
Compositional and density differences between melt and solids might have
complex feedbacks with the interior fluid dynamics of lava planets. For
instance, liquid silicates might become denser than their solids counterparts
at high pressures (Funamori & Sato, 2010; Boukaré et al., 2015; Boukaré &
Ricard, 2017; Caracas et al., 2019). Compared to previous studies that suggest
a shallow magma ocean, our modeling points towards a deep hemispherical magma
ocean that is expected to have profound implications for the atmosphere.
Indeed, the composition of the atmosphere is dictated by the replenishment of
surface material with fertile material coming from the magma ocean.

Canonical magma ocean dynamics do not readily apply to tidally locked planets.
The usual Rayleigh-Bénard convection in magma oceans, solid mantles and liquid
cores develops in a fluid layer that is either cooled from above or heated
from below (e.g., Solomatov, 2000). In this regime, heat transport occurs
vertically. For tidally locked lava planets, the temperature difference
between the two hemispheres is expected to drive horizontal convection (e.g.,
Hughes & Griffiths, 2008). In this regime, heat transport proceeds
horizontally similar to the thermohaline circulation in the Earth’s oceans and
the vertical temperature gradients is expected to differ from an adiabat.
Indeed, thermal forcing at the surface results in thermal stratification in
the bulk fluid, making the temperature profile subadiabatic in the dayside
magma and superadiabtic on the night side.

The high surface temperature of a lava planet does not necessary preclude the
possibility of Rayleigh-Bénard-like convection. Indeed, the base of the rocky
mantle is likely hotter the surface due to the slowly cooling iron core. Two
scenarios could be envisioned in future studies depending on whether basal
heating from the core is stronger compared to surface heating on the day-side.
If basal heating is neglible compared to surface heating, the efficient
cooling by advection on the night side may maintain a cold and solidified
interior, on average. In that case, the present study probably overestimates
the depth of the magma ocean as the interior temperature is not expected to
follow an adiabat. We do not favor this hypothesis, however, as it appears
unlikely that the temperature at the core-mantle boundary is lower than 3000
K. The surface temperature of the Earth is approximately 300 K and its CMB
temperature is higher than 4000 K (e.g., Andrault et al., 2011; Fiquet et al.,
2010; Labrosse, 2015).

The maximum incoming heat flux on the day-side can be approximated by a
conductive heat flux across a layer the size of the mantle. Using a thermal
conductivity k = 1 W m-1 K-1, a temperature difference $\Delta T$ = 3000 K and
mantle thickness H = 1.5 $R_{\oplus}$ /2 , we get a surface heat flux of about
0.3 TW. This surface heat flux is probably negligible compared to the secular
cooling of the core that is of the order of 10 TW (today) to 60 TW (4 Gy ago)
for the Earth (e.g., Labrosse, 2015; Nimmo & Schubert, 2015). Consequently,
basal heating of the mantle is expected to drive Rayleigh-Bénard convection
and buffer the temperature profile to an adiabat, keeping the interior warm
and molten on the day side.

Our calculations can be considered as a first order estimation that must be
complemented by geodynamic models. However, this study raises interesting
questions regarding lava planet’s interior dynamics. Fractionation by liquid-
gas equilibrium at the surface has been proposed in several studies (e.g.,
Schaefer & Fegley, 2009; Kite et al., 2016). Here, we emphasize the role of
solid-liquid fractionation in the planet’s interior. How do convection
dynamics transition from the molten region to the solid hemisphere? How do
compositional differences between melt and solid affect the interior dynamics
? How does magma ocean composition evolve with time? How do mixing and
fractionation compete to determine the steady-state compositon of the magma
ocean? Since the atmosphere forms in equilibrium with the magma, any processes
that control its composition at the surface are crucial for the atmosphere.
Therefore, magma ocean interior dynamics must be considered as a fundamental
aspect of lava planets atmospheres evolution.

###  4.2 Implications for (incompatible) volatile elements

Models of magma oceans generally consider liquid–gas equilibrium at the planet
surface as a primary mechanism for atmosphere formation (e.g., Abe & Matsui,
1986; Bower et al., 2021). However, vapor can also degas from the melt at
depth when it becomes supersaturated in volatile, as observed in terrestrial
volcanoes (e.g., Le Guern et al., 1979; Tazieff, 1994). This process might be
an alternative to surface degassing when the magma ocean is isolated from the
atmosphere by a solid conductive lid (e.g., Bower et al., 2021) and thus
reinforces the efficiency of a lava planet to degas its volatile content in
regions where the surface temperature is below the solidus, i.e., near the
terminator and on the night-side.

In the context of a rocky atmosphere, the fate of this putative volatile
atmosphere is however uncertain. We expect strong winds from the day side to
the night side. Studies of lava planet atmospheres have focused on Na because
of its relative volatility (Castan & Menou, 2011; Kite et al., 2016; Nguyen et
al., 2020) or SiO because of its abundance in the mantle (Nguyen et al., 2020;
Zieba et al., 2022). In either case, the atmosphere is expected to flow
towards the day —night terminator at velocities of kilometers per second,
often exceeding the speed of sound. It has generally been assumed that any
water present at the surface of the planet would be lost to space or cold
trapped on the perpetually dark regions of the planet. It is possible,
however, that the gradual resurfacing of the nightside due to mantle overturn
could resupply water to the magma ocean. In that case, we predict that water
would be degassed near the day—night terminator, only to be swept to the
dayside of the planet by the background SiO wind. The presence of volatile
species near the planet terminator could be tested in the near future with
transit spectroscopy.

##  5 Conclusion

We have used state-of-the-art 1D models to make first-order predictions for
the magma ocean on a lava planet. While these models are standard for Solar
System studies, they have not yet been widely adopted in the context of
exoplanets. The key to our results is the concept of partial melt and the high
pressure thermodynamics of melting curves.

A rocky planet’s mantle is a multi-component system and hence has a gradual
transition from liquid to solid rather than the sharp phase transitions of
pure substances such as water. As a result, a lava planet has no well-defined
shore but rather a swath thousands of km wide where magma becomes increasingly
sluggish.

The partial melting is also thermodynmically important: latent heat from the
phase transition ensures that the isentrope is closer to isothermal and hence
the transition to solid, or even sluggish liquid, occurs at higher pressures.
We therefore predict that the magma ocean on K2-141b extends down to the
core–mantle boundary.

Partial crystallization near the edge of the magma ocean leads to gradients in
the surface composition of the melt and hence may affect the composition of
the overlying rock vapor atmosphere. Near the mush–solid interface the melt
becomes saturated in water, resulting in exsolution of water near the
day–night terminator. Future work is needed to predict whether this could be a
steady-state phenomenon or whether the water becomes permanently trapped as
ice sheets on the planet’s nightside.

Finally, we have adopted isentropic vertical temperature profiles in the magma
ocean but the substantial horizontal gradients in temperature, chemistry,
evaporation and precipitation likely leads to magma ocean dynamics completely
different from those considered in the Solar System context. Future challenges
include numerically simulating the dynamics of lava planet magma oceans, and
coupling such simulations to the atmospheric chemistry and dynamics that are
probed by astronomical observations.

## Acknoledgement

We thank the anonymous reviewer for his/her insightful comments that helped
improve the paper. This work has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program (grant agreement no. 101019965— SEPtiM). Parts of this work were
supported by the UnivEarthS Labex program at Université de Paris and IPGP
(ANR-10-LABX-0023 and ANR-11-IDEX-0005-02). We acknowledge the camaraderie and
support of the McGill Space Institute and the ”Institut de Recherche sur les
exoplanètes” (IREX).

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# The Borel Ramsey Properties for Countable Borel Equivalence Relations

Su Gao  School of Mathematical Sciences and LPMC, Nankai University, Tianjin
300071, P.R. China<EMAIL_ADDRESS>and  Ming Xiao  School of Mathematical
Sciences and LPMC, Nankai University, Tianjin 300071, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

We define some natural notions of strong and weak Borel Ramsey properties for
countable Borel equivalence relations and show that they hold for a countable
Borel equivalence relation if and only if the equivalence relation is smooth.
We also consider some variation of the notion for hyperfinite non-smooth Borel
equivalence relations.

###### 2020 Mathematics Subject Classification:

Primary 03E15; Secondary 05C55

The authors acknowledge the partial support of their research by the National
Natural Science Foundation of China (NSFC) grants 12250710128 and 12271263.

##  1\. Introduction

For an integer $n>0$ and a set $X$, let $[X]^{n}$ denote the set of all
$n$-element subsets of $X$. The celebrated infinite Ramsey theorem, proved by
Ramsey in his seminal paper [6], is a cornerstone of infinite combinatorics.

######  Theorem 1.1 (Ramsey).

For any integer $n,k>0$ and function $c:[\mathbb{N}]^{n}\to\\{1,\dots,k\\}$,
there exists $a\in\\{1,\dots,k\\}$ and an infinite subset
$M\subseteq\mathbb{N}$ such that for all $x\in[M]^{n}$, $c(x)=a$.

Here we view $[\mathbb{N}]^{n}$ as a hypergraph on the set $\mathbb{N}$, which
we call the complete hypergraph of dimension $n$ over $\mathbb{N}$; we regard
$c$ as a coloring of the hyperedges by $k$ many colors, and $M$ is often
referred to as a monochromatic subset. In the arrow notation of Erdős and Rado
[2], the infinite Ramsey theorem can be abbreviated as
$\aleph_{0}\rightarrow(\aleph_{0})^{n}_{k}$.

In this paper, we study generalizations of the infinite Ramsey theorem in the
context of invariant descriptive set theory. Specifically, we consider
invariant versions of Ramsey-type properties for aperiodic countable
equivalence relations. We also impose definability conditions on the
equivalence relation, the coloring and the monochromatic set. For basic
results of invariant descriptive set theory and undefined terminology in this
paper, the reader can consult [4] or [3].

In our setting, let $X$ be a standard Borel space and let $E$ be an
equivalence relation on $X$. Recall that $E$ is called a Borel equivalence
relation if $E$ is a Borel subset of $X^{2}$, $E$ is countable if each
equivalence class of $E$ is countable, and $E$ is aperiodic if each
equivalence class of $E$ is infinite. Thus, if $E$ is an aperiodic, countable
Borel equivalence relation on a Polish space $X$, the quotient space $X/E$ can
be viewed as a uniform collection of countably infinite sets on which the
Ramsey-type properties can be studied. If the Axiom of Choice is employed,
then the infinite Ramsey theorem for this invariant context follows
immediately from the classical infinite Ramsey theorem. As usual, if we
require that the monochromatic subset be Borel, then the invariant version of
the infinite Ramsey theorem can fail. The main objective of this paper is to
characterize exactly when these Ramsey-type properties continue to hold in the
invariant Borel context.

Let $E$ be an aperiodic, countable Borel equivalence relation on a standard
Borel space $X$. For an integer $n>0$, let

$[X]^{n}_{E}=\left\\{A\in[X]^{n}\colon\exists x\in X\
A\subseteq[x]_{E}\right\\},$

where $[x]_{E}$ denotes the $E$-equivalence class of $x\in X$. $[X]^{n}_{E}$
is a standard Borel space. For an integer $k>0$, we call a Borel function
$c:[X]^{n}_{E}\to\\{1,\dots,k\\}$ a Borel $E$-coloring of dimension $n$ over
$X$ by $k$ many colors.

Recall a subset $Y\subseteq X$ is called a complete section for $E$ if for all
$x\in X$, $Y\cap[x]_{E}\neq\varnothing$. A complete section $Y$ for $E$ is
infinite if its intersection with every $E$-equivalence class is infinite.
Given a Borel $E$-coloring of dimension $n$ over $X$ by $k$ many colors, a
complete section $Y\subseteq X$ is $E$-monochromatic if for every $x\in X$,
there is $a\in\\{1,\dots,k\\}$ such that for all $A\in[Y]^{n}$ with
$A\subseteq[x]_{E}$, $c(A)=a$.

We define the following Ramsey-type properties in the invariant Borel context.

######  Definition 1.2.

Let $X$ be a standard Borel space and let $E$ be an aperiodic, countable Borel
equivalence relation on $X$. For integers $n,k>0$, the (strong) Borel Ramsey
property $E\rightarrow_{B}(E)^{n}_{k}$ means that for any Borel $E$-coloring
$c$ of dimension $n$ over $X$ by $k$ many colors, there exists a Borel
$E$-monochromatic, infinite complete section $Y\subseteq X$. The weak Borel
Ramsey property $E\rightarrow_{B}^{*}(E)^{n}_{k}$ means that for any Borel
$E$-coloring $c$ of dimension $n$ over $X$ by $k$ many colors, there exists a
Borel $E$-monochromatic complete section.

It is easy to see that if $n=1$ or $k=1$, then the Ramsey properties hold for
all aperiodic, countable Borel equivalence relations $E$ on a standard Borel
space. For $n,k>1$, it turns out that these notions are closely related to the
concept of smoothness in the Borel reducibility hierarchy. Recall that for
equivalence relations $E$ and $F$ on standard Borel spaces $X$ and $Y$
respectively, we say that $E$ is Borel reducible to $F$, denoted $E\leq_{B}F$,
if there is a Borel function $f:X\to Y$ such that for all $x_{1},x_{2}\in X$,

$x_{1}Ex_{2}\iff f(x_{1})Ff(x_{2}).$

$E$ is smooth if it is Borel reducible to the equality relation on a standard
Borel space, i.e., there is a standard Borel space $Y$ such that $E\leq_{B}\
=\upharpoonright\\!Y$.

Our main theorem is the following.

######  Theorem 1.3.

Let $X$ be a standard Borel space and let $E$ be an aperiodic, countable Borel
equivalence relation on $X$. The following are equivalent:

  1. (i)

The Borel Ramsey property $E\rightarrow_{B}(E)^{n}_{k}$ holds for some
integers $n,k>1$;

  2. (ii)

The Borel Ramsey property $E\rightarrow_{B}(E)^{n}_{k}$ holds for all integers
$n,k>1$;

  3. (iii)

The weak Borel Ramsey property $E\rightarrow^{*}_{B}(E)^{n}_{k}$ holds for
some integers $n,k>1$;

  4. (iv)

The weak Borel Ramsey property $E\rightarrow^{*}_{B}(E)^{n}_{k}$ holds for all
integers $n,k>1$;

  5. (v)

$E$ is smooth.

##  2\. Proof of the Main Theorem

In this section we prove Theorem 1.3. We first note that the strong and weak
Borel Ramsey properties are equivalent.

######  Lemma 2.1.

Let $X$ be a standard Borel space and $E$ be an aperiodic, countable Borel
equivalence relation on $X$. For any integers $n,k>1$, if
$E\rightarrow^{*}_{B}(E)^{n}_{k}$ then $E\rightarrow_{B}(E)^{n}_{k}$.

###### Proof.

Let $c:[X]^{n}_{E}\to\\{1,\dots,k\\}$ be a Borel $E$-coloring. Let $Y\subseteq
X$ be a Borel $E$-monochromatic complete section. Let

$Y^{\prime}=\\{y\in Y\,:\,\mbox{$[y]_{E}\cap Y$ is finite}\\}.$

Then $Y^{\prime}$ is Borel. If $Y^{\prime}=\varnothing$ then $Y$ is an
infinite complete section and there is nothing to prove. Thus we assume
$Y^{\prime}\neq\varnothing$. Let $X^{\prime}$ be the $E$-saturation of
$Y^{\prime}$. Then $X^{\prime}$ is a standard Borel space, and $Y^{\prime}$ is
a Borel complete section of $E\\!\upharpoonright\\!X^{\prime}$. To prove the
lemma it suffices to find a Borel
$E\\!\upharpoonright\\!X^{\prime}$-monochromatic, infinite complete section
$Z$. For notational simplicity, assume $Y^{\prime}=Y$ and $X^{\prime}=X$,
i.e., each $E\\!\upharpoonright\\!Y$-class is finite. Furthermore, let $<$ be
a Borel linear ordering of $X$. By picking the $<$-least element from each
$E\\!\upharpoonright\\!Y$-class, we may assume that $Y$ is a Borel transversal
for $E$, i.e., for each $x\in X$, $[x]_{E}\cap Y$ is a singleton. Let
$\sigma:X\to Y$ be the function with the graph $\\{(x,y)\,:\,x\in X,y\in
Y,xEy\\}$. Then $\sigma$ is a Borel selector for $E$.

By the Feldman–Moore theorem (c.f., e.g., [3, Theorem 7.1.4]), there is a
Borel action of a countable group $\Gamma$ on $X$ which generates $E$. Let
$\\{g_{i}\\}_{i\in\mathbb{N}}$ enumerate the elements of $\Gamma$, with
$g_{0}=1_{\Gamma}$. We define a wellordering $\prec$ of each $E$-class of $X$
as follows. For $xEx^{\prime}$, define

$x\prec x^{\prime}\iff\exists i\ \left[\,g_{i}\cdot\sigma(x)=x\wedge\forall
j\leq i\ g_{j}\cdot\sigma(x)\neq x^{\prime}\,\right].$

$\prec$ is clearly Borel. For each $y\in Y$, $\prec$ wellorders $[y]_{E}$,
with $y$ as the $\prec$-least element of $[y]_{E}$.

We then proceed as in the proof of the classical infinite Ramsey theorem for
each $[y]_{E}$, and observe that all quantifiers involved are first-order with
respect to natural numbers and so the resulting set is Borel.

The proof is by induction on $n$. If $n=1$, we use the usual pigeonhole
principle to pick an infinite monochromatic subset of $[y]_{E}$, whereas the
wellordering $\prec$ helps to guarantee that we can pick the elements in a
Borel manner.

Now suppose that $c$ is of dimension $n>1$ and for every standard Borel space
$X^{\prime}$, for every aperiodic, countable Borel equivalence relation
$E^{\prime}$ on $X^{\prime}$, and for every Borel $E^{\prime}$-coloring
$c^{\prime}$ of dimension $n-1$ over $X^{\prime}$, there is a Borel
$E^{\prime}$-monochromatic, infinite complete section for $c^{\prime}$. Let
$X_{0}=X$ and $Y_{0}=Y$. Let $c_{0}$ be the Borel $E$-coloring of dimension
$n-1$ over $X\setminus Y$ by $k$ many colors defined by

$c_{0}(\\{x_{1},...,x_{n-1}\\})=c(\\{\sigma(x_{1}),x_{1},...,x_{n-1}\\}).$

Let $X_{1}\subseteq X\setminus Y$ be a Borel $E$-monochromatic, infinite
complete section for $c_{0}$. Let

$Y_{1}=\\{x\in X_{1}\,:\,\mbox{$x$ is the $\prec$-least element of
$[x]_{E}\cap X_{1}$}\\}$

and

$c_{1}(\\{x_{1},\dots,x_{n-1}\\})=c(\\{y,x_{1},\dots,x_{n-1}\\})$

for $\\{x_{1},\dots,x_{n-1}\\}\in[X_{1}\setminus Y_{1}]^{n-1}_{E}$, where
$y\in Y_{1}$ is the unique element with $yEx_{1}$.

Repeating this process, we obtain infinite sequences
$\\{X_{j}\\}_{j\in\mathbb{N}},\\{Y_{j}\\}_{j\in\mathbb{N}},\\{c_{j}\\}_{j\in\mathbb{N}}$
so that the following hold for all $j\in\mathbb{N}$:

  1. (1)

$Y_{j}\subseteq X_{j}$ is a Borel transversal of $E$ on $X$;

  2. (2)

if $y\in Y_{j}$, $x\in X_{j}$ and $xEy$, then $y\prec x$;

  3. (3)

$c_{j}$ is a Borel $E$-coloring of dimension $n-1$ over $X_{j}\setminus Y_{j}$
by $k$ many colors;

  4. (4)

if $y\in Y_{j}$, $\\{x_{1},\dots,x_{n-1}\\}\in[X_{j}\setminus
Y_{j}]^{n-1}_{E}$ and $yEx_{1}$, then

$c_{j}(\\{x_{1},...,x_{n-1}\\})=c(\\{y,x_{1},...,x_{n-1}\\});$

  5. (5)

$X_{j+1}\subseteq X_{j}\setminus Y_{j}$ is a Borel $E$-monochromatic, infinite
complete section for $c_{j}$;

  6. (6)

if $j^{\prime}<j$, $y^{\prime}\in Y_{j^{\prime}}$, $y\in Y_{j}$ and
$y^{\prime}Ey$, then $y^{\prime}\prec y$.

Let $Y_{\infty}=\bigcup_{j\in\mathbb{N}}Y_{j}$. For each fixed $y\in
Y_{\infty}$, consider $Y_{y}=\\{z\in Y_{\infty}\,:\,y\prec z\\}$. If we define
$c_{y}:[Y_{y}]^{n-1}\to\\{1,\dots,k\\}$ by

$c_{y}(\\{x_{1},\dots,x_{n-1}\\})=c(\\{y,x_{1},\dots,x_{n-1}\\}),$

then $c_{y}$ is constant on $[Y_{y}]^{n-1}$. Let $c^{\prime\prime}(y)$ be this
constant. Then $c^{\prime\prime}$ is a Borel $E$-coloring of dimension $1$
over $Y_{\infty}$ by $k$ many colors. By the inductive hypothesis for $n=1$,
we obtain a Borel $E$-monochromatic, infinite complete section $Z\subseteq
Y_{\infty}$ for $c^{\prime\prime}$. One can readily check that $Z$ is in fact
a Borel $E$-monochromatic, infinite complete section for $c$. ∎

As a corollary of the above proof, we obtain a half of our main theorem.

######  Proposition 2.2.

Let $X$ be a standard Borel space and $E$ be an aperiodic, countable Borel
equivalence relation on $X$. If $E$ is smooth, then for any integers $n,k>1$,
the Borel Ramsey property $E\rightarrow_{B}(E)^{n}_{k}$ holds.

###### Proof.

By a theorem of Kechris (c.f., e.g. [3, Theorem 5.4.11]), any smooth,
countable Borel equivalence relation has a Borel transversal. The proof of
Lemma 2.1 shows that $E\rightarrow_{B}(E)^{n}_{k}$ holds. ∎

In view of Lemma 2.1 we will not distinguish between the properties
$E\rightarrow_{B}(E)^{n}_{k}$ and $E\rightarrow^{*}_{B}(E)^{n}_{k}$ in the
remaining discussions. Next we note that the Borel Ramsey properties are
preserved by infinite complete sections.

######  Lemma 2.3.

Let $X$ be a standard Borel space and $E$ be an aperiodic, countable Borel
equivalence relation on $X$. Let $n,k>1$ be integers. Then the following are
equivalent:

  1. (i)

The Borel Ramsey property $E\rightarrow_{B}(E)^{n}_{k}$ holds;

  2. (ii)

For every Borel infinite complete section $Y\subseteq X$,
$F\rightarrow_{B}(F)^{n}_{k}$ holds, where $F=E\\!\upharpoonright\\!Y$;

  3. (iii)

There is a Borel infinite complete section $Y\subseteq X$ so that
$F\rightarrow_{B}(F)^{n}_{k}$ holds, where $F=E\\!\upharpoonright\\!Y$.

###### Proof.

The direction (ii)$\Rightarrow$(iii) is trivial. To see (iii)$\Rightarrow$(i)
it suffices to note that a Borel infinite complete section of a Borel infinite
complete section is still a Borel infinite complete section.

Next we prove (i)$\Rightarrow$(ii). Again, by the Feldman–Moore theorem, there
is a Borel action of a countable group $\Gamma$ on $X$ which generates $E$.
Let $\\{g_{i}\\}_{i\in\mathbb{N}}$ enumerate the elements of $\Gamma$, with
$g_{0}=1_{\Gamma}$. Let $Y\subseteq X$ be a Borel infinite complete section
and let $F=E\\!\upharpoonright\\!Y$. For each $x\in X$, let
$i_{Y}(x)=\min\\{i\colon g_{i}\cdot x\in Y\\}$. Write
$g_{Y}(x)=g_{i_{Y}(x)}(x)$. So $g_{Y}$ is a Borel $E$-invariant function with
range in $Y$ and is identity on $Y$.

Fix a Borel $F$-coloring $c:[Y]_{F}^{n}\to\\{1,\dots,k\\}$. In view of Lemma
2.1, it suffices to find a Borel $F$-monochromatic complete section
$Z\subseteq Y$. For this, we extend $c$ to $c^{\prime}$ on $[X]^{n}_{E}$ by
defining

$c^{\prime}(\\{x_{1},\dots,x_{n}\\})=\left\\{\begin{array}[]{ll}c(\\{g_{Y}(x_{1}),...,g_{Y}(x_{n})\\})&\mbox{if
$\\{g_{Y}(x_{1}),\dots,g_{Y}(x_{n})\\}\in[Y]^{n}$,}\\\
1&\mbox{otherwise.}\end{array}\right.$

This is clearly a Borel $E$-coloring. By (i), there is a Borel
$E$-monochromatic, infinite complete section $W$ for $c^{\prime}$. Let
$Z=\\{g_{Y}(x)\colon x\in W\\}$. Clearly $Z$ is a Borel complete section of
$Y$. To see that $Z$ is $E$-monochromatic, let
$\\{z_{1},\dots,z_{n}\\}\in[Z]^{n}_{E}$. There are $x_{1},\dots,x_{n}\in W$ so
that $g_{Y}(x_{j})=z_{j}$ for every $1\leq j\leq n$. By definition,

$c^{\prime}(\\{x_{1},\dots,x_{n}\\})=c(\\{g_{Y}(x_{1}),...,g_{Y}(x_{n})\\})=c(\\{z_{1},\dots,z_{n}\\}).$

Thus $Z$ is $E$-monochromatic. ∎

A further consequence is the preservation of the Borel Ramsey properties by
Borel subsets.

######  Lemma 2.4.

Let $X$ be a standard Borel space and $E$ be an aperiodic, countable Borel
equivalence relation on $X$. Let $n,k>1$ be integers. Let $Y\subseteq X$ be
Borel and let $F=E\\!\upharpoonright\\!Y$. Assume $F$ is aperiodic. If
$E\rightarrow_{B}(E)^{n}_{k}$, then $F\rightarrow_{B}(F)^{n}_{k}$.

###### Proof.

Let $X^{\prime}$ be the $E$-saturation of $Y$ and
$E^{\prime}=E\\!\upharpoonright\\!X^{\prime}$. Then $X^{\prime}$ is a standard
Borel space and $E^{\prime}$ is an aperiodic, countable Borel equivalence
relation on $X^{\prime}$. It is easy to see that
$E^{\prime}\rightarrow_{B}(E^{\prime})^{n}_{k}$ follows from
$E\rightarrow_{B}(E)^{n}_{k}$. Now $Y$ is a Borel infinite complete section of
$X^{\prime}$, and hence $F\rightarrow_{B}(F)^{n}_{k}$ by Lemma 2.3. ∎

To state the next result we recall the notion of Borel embedding between
equivalence relations. Let $E$ and $F$ be equivalence relations on standard
Borel spaces $X$ and $Y$ respectively. We say that $E$ is Borel embeddable
into $F$, denoted $E\sqsubseteq_{B}F$, if there is an injective Borel function
$f:X\to Y$ such that for all $x_{1},x_{2}\in X$, $x_{1}Ex_{2}$ iff
$f(x_{1})Ff(x_{2})$.

######  Lemma 2.5.

Let $X,Y$ be standard Borel spaces and let $E,F$ be aperiodic, countable Borel
equivalence relations on $X,Y$, respectively. Let $n,k>1$ be integers. If
$E\sqsubseteq_{B}F$ and $F\rightarrow_{B}(F)^{n}_{k}$, then
$E\rightarrow_{B}(E)^{n}_{k}$.

###### Proof.

Let $f(X)\subseteq Y$ be the image of $X$ under $f$. Then $f(X)$ is Borel by a
theorem of Luzin–Suslin (c.f., e.g. [4, Theorem 15.1]). Let
$F^{\prime}=F\\!\upharpoonright\\!f(X)$. Then $F^{\prime}$ is aperiodic. By
Lemma 2.4, $F^{\prime}\rightarrow_{B}(F^{\prime})^{n}_{k}$ holds.

Let $c$ be a Borel $E$-coloring of dimension $n$ over $X$ by $k$ many colors.
Push forward $c$ to an $F^{\prime}$-coloring $c^{\prime}$ over $f(X)$ by
defining

$c^{\prime}(\\{f(x_{1}),\dots,f(x_{n})\\})=c(\\{x_{1},\dots,x_{n}\\}).$

$c^{\prime}$ is well defined by the injectivity of $f$. Let $Z$ be a Borel
$F^{\prime}$-monochromatic complete section for $c^{\prime}$. Then $f^{-1}(Z)$
is a Borel $E$-monochromatic complete section for $c$. ∎

We are now ready to prove Theorem 1.3. In our proof we will need to study the
equivalence relation $E_{0}$ defined on the Cantor space
$2^{\mathbb{N}}=\\{0,1\\}^{\mathbb{N}}$:

$xE_{0}y\iff\exists n\ \forall m\geq n\ x(m)=y(m).$

Let $\Gamma_{0}$ be the collection of all finite subsets of $\mathbb{N}$. For
$g,h\in\Gamma_{0}$, define the multiplication operation by $gh=g\triangle h$,
i.e. the symmetric difference of $g$ and $h$. Then $\Gamma_{0}$ is a countable
group. For $g\in\Gamma_{0}$ and $x\in 2^{\mathbb{N}}$, define

$(g\cdot x)(n)=\left\\{\begin{array}[]{ll}x(n)&\mbox{if $n\not\in g$,}\\\
1-x(n)&\mbox{otherwise, i.e., if $n\in g$.}\end{array}\right.$

This defines a continuous action of $\Gamma_{0}$ on $2^{\mathbb{N}}$ which
generates $E_{0}$. By a theorem of Harrington–Kechris–Louveau, better known as
the Glimm-Effros dichotomy theorem (c.f., e.g. [3, Theorem 6.3.1]), for any
Borel equivalence relation $E$ on a standard Borel space $X$, either $E$ is
smooth or else $E_{0}\sqsubseteq_{B}E$. In particular, $E_{0}$ is not smooth.

###### Proof of Theorem 1.3.

In view of Lemma 2.1 and Proposition 2.2, we only need to verify the direction
(i)$\Rightarrow$(v). For this we first construct a Borel $E_{0}$-coloring of
dimension $2$ over $2^{\mathbb{N}}$ by $2$ colors that has no Borel
$E_{0}$-monochromatic, infinite complete section.

For any distinct $x,y\in 2^{\mathbb{N}}$ with $xE_{0}y$, let

$\delta(x,y)=\min\\{n\colon\forall m\geq n\ x(m)=y(m)\\}$

and define

$c(\\{x,y\\})=\left\\{\begin{array}[]{ll}1&\mbox{if
$\sum_{i<\delta(x,y)}x(i)+\sum_{i<\delta(x,y)}y(i)$ is even,}\\\ \\\
2&\mbox{otherwise, i.e., if
$\sum_{i<\delta(x,y)}x(i)+\sum_{i<\delta(x,y)}y(i)$ is
odd.}\end{array}\right.$

We claim that there cannot be a Borel $E_{0}$-monochromatic, infinite complete
section for $c$. To see this, we define a new equivalence relation $F$ which
refines $E_{0}$ by

$xFy\iff x=y\mbox{ or }c(\\{x,y\\})=1.$

It is straightforward to check that this is indeed an equivalence relation. It
is also easy to verify that each $E_{0}$-class consists of exactly two
$F$-classes.

In fact, let $\Gamma_{0}$ be the group defined before this proof. Let
$\Gamma_{1}$ be the subgroup of $\Gamma_{0}$ that consists of only
$g\in\Gamma_{0}$ with $|g|$ even. Then $\Gamma_{1}$ has index $2$ in
$\Gamma_{0}$, and $F$ is generated by the continuous action of $\Gamma_{1}$ on
$2^{\mathbb{N}}$ as a subaction of $\Gamma_{0}$.

Now assume $Y$ is a Borel $E_{0}$-monochromatic, infinite complete section.
Note that if $c$ is constant on a subset of a single $E_{0}$-class and the
constant value is $2$, then this subset must have size at most $2$. It follows
that on $[Y]^{2}_{E_{0}}$, $c$ always takes the value $1$. This means that
$E_{0}\\!\upharpoonright\\!Y=F\\!\upharpoonright\\!Y$.

Let $W$ be the $F$-saturation of $Y$. Then $W$ is Borel and $F$-invariant. In
addition, for $g_{0}=\\{0\\}$, we have that $W\cup g_{0}\cdot
W=2^{\mathbb{N}}$ and $W\cap g_{0}\cdot W=\varnothing$. However, by the first
topological 0-1 law (c.f. [4, Theorem 8.46]), $W$ is either meager or comeager
as a subset of $2^{\mathbb{N}}$. This is a contradiction. We have shown
$E_{0}\not\rightarrow_{B}(E_{0})^{2}_{2}$.

Next we show that $E_{0}\not\rightarrow_{B}(E_{0})^{n}_{k}$ for any $n,k>1$.
For this, fix integers $n,k>1$. Let $<$ be a Borel linear ordering of
$2^{\mathbb{N}}$. If $x_{1},\dots,x_{n}\in 2^{\mathbb{N}}$, $x_{1}E\dots
Ex_{n}$ and $x_{1}<\cdots<x_{n}$, then we define

$c_{n}(\\{x_{1},\dots,x_{n}\\})=c(\\{x_{1},x_{2}\\}).$

$c_{n}$ is a Borel $E_{0}$-coloring of dimension $n$ over $2^{\mathbb{N}}$ by
$k$ many colors (in fact only 2 colors are used). If $Z$ was a Borel
$E_{0}$-monochromatic, infinite complete section for $c_{n}$, then it would
also be a Borel $E_{0}$-monochromatic, infinite complete section for $c$. Thus
$E_{0}\not\rightarrow_{B}(E_{0})^{n}_{k}$.

By the Glimm–Effros dichotomy theorem and Lemma 2.5, if $E$ is nonsmooth, then
$E_{0}\sqsubseteq_{B}E$ and therefore $E\not\rightarrow(E)^{n}_{k}$ for any
$n,k>1$. This is the contrapositive of (i)$\Rightarrow$(v). ∎

##  3\. Hyperfinite Nonsmooth Equivalence Relations

Throughout this section we fix a standard Borel space $X$ and an aperiodic,
countable Borel equivalence relation $E$ on $X$. Also fix integers $n,k>1$. As
a consequence of our main theorem, the Borel Ramsey property
$E\rightarrow_{B}(E)^{n}_{k}$ can be restated as: for any Borel $E$-coloring
$c:[X]^{n}_{E}\to\\{1,\dots,k\\}$, there exists a Borel function $S:X\to
X^{\mathbb{N}}$ such that

  1. (a)

for any $x\in X$ and distinct $i,j\in\mathbb{N}$, $S(x)(i)\neq S(x)(j)$;

  2. (b)

for any $x\in X$, the set $\\{S(x)(i)\colon i\in\mathbb{N}\\}$ is a
monochromatic subset of $[x]_{E}$;

  3. (c)

for any $x,y\in X$, $xEy$ iff $S(x)=S(y)$.

This restatement makes it clear that the Borel Ramsey property implies the
smoothness of $E$. If we replace (c) by

  1. (c’)

for any $x,y\in X$, if $xEy$ then $\\{S(x)(i)\colon
i\in\mathbb{N}\\}=\\{S(y)(i)\colon i\in\mathbb{N}\\}$,

the two statements are still equivalent. Thus, if we want to consider a
meaningful notion of Ramsey-type property for nonsmooth equivalence relations,
we have to allow the sets $\\{S(x)(i)\colon i\in\mathbb{N}\\}$ and
$\\{S(y)(i)\colon i\in\mathbb{N}\\}$ to differ.

In this section we consider hyperfinite nonsmooth equivalence relations.
Recall that $E$ is finite if every $E$-class is finite, and $E$ is hyperfinite
if there is an increasing sequence of Borel finite equivalence relations
$\\{F_{i}\\}_{i\in\mathbb{N}}$ such that $E=\bigcup_{i}F_{i}$. Thus,
hyperfinite equivalence relations are necessarily countable Borel equivalence
relations. By a theorem of Dougherty–Jackson–Kechris (c.f. [1] or [3, Theorem
7.2.3]), $E$ is hyperfinite iff $E\leq_{B}E_{0}$ iff there is a Borel
assigment $C\mapsto<_{C}$ associating with each $E$-class $C$ a linear
ordering $<_{C}$ of $C$ so that there is an order-preserving map from
$(C,<_{C})$ into $(\mathbb{Z},<)$. For an aperiodic hyperfinite equivalence
relation $E$ we may require that the order type of $(C,<_{C})$ is $\zeta$, the
order type of $(\mathbb{Z},<)$, i.e., there is an order isomorphism between
$(C,<_{C})$ and $(\mathbb{Z},<)$.

An example of a nonsmooth Borel equivalence relation which is closely related
to $E_{0}$ is the equivalence relation $E_{0}(X)$, defined on $X^{\mathbb{N}}$
for any standard Borel space $X$ by

$xE_{0}(X)y\iff\exists i\ \forall j\geq i\ x(j)=y(j).$

When $X$ is countable, we still have $E_{0}(X)\leq_{B}E_{0}$ and thus
$E_{0}(X)$ is hyperfinite. When $X$ is uncountable, $E_{0}(X)$ is no longer a
countable Borel equivalence relation, but by another theorem of
Dougherty–Jackson–Kechris (c.f. [1] or [3, Theorem 8.1.5]), for any countable
Borel equivalence relation $E$, if $E\leq_{B}E_{0}(X)$, then $E$ is
hyperfinite. Following [5], we denote $E_{0}(X)$ by $E_{1}$ when $X$ is an
uncountable standard Borel space.

To formulate an appropriate Ramsey-type property for hyperfinite nonsmooth
equivalence relations, a natural idea is to allow a small (finite) difference
between $S(x)$ and $S(y)$ for $xEy$. This is stated precisely in the following
definition.

######  Definition 3.1.

Given an $E$-coloring $c:[X]^{n}_{E}\to\\{1,\dots,k\\}$. A function $S:X\to
X^{\mathbb{N}}$ is a monochromatic reduction to $E_{1}$ for $c$ if:

  1. (1)

for any $x\in X$ and distinct $i,j\in\mathbb{N}$, $S(x)(i)\neq S(y)(j)$;

  2. (2)

for any $x\in X$, the set $\\{S(x)(i)\colon i\in\mathbb{N}\\}$ is a
monochromatic subset of $[x]_{E}$;

  3. (3)

for any $x,y\in X$, $xEy$ iff $S(x)E_{1}S(y)$.

However, we note the following negative result.

######  Lemma 3.2.

There is a Borel $E_{0}$-coloring $c$ of dimension $2$ by $2$ colors such that
there exists no Borel monochromatic reduction to $E_{1}$ for $c$.

###### Proof.

Consider the $E_{0}$-coloring $c$ defined in the proof of Theorem 1.3. Let $F$
be the subequivalence relation of $E_{0}$ also defined in that proof. Assume
$S$ is a Borel monochromatic reduction to $E_{t}$ for $c$. Then for each $x$,
$\\{S(x)(i)\colon i\in\mathbb{N}\\}$ is an infinite monochromatic subset of
$[x]_{E_{0}}$, hence it is included in a single $F$-class. If $xE_{0}y$, then
$S(x)E_{1}S(y)$, and it follows that $\\{S(x)(i)\colon i\in\mathbb{N}\\}$ and
$\\{S(y)(i)\colon i\in\mathbb{N}\\}$ are included in a single $F$-class. Now
let $Y=\\{S(x)(i)\,:\,i\in\mathbb{N},\ x\in X\\}$. Then $Y$ is a Borel
$E_{0}$-monochromatic, infinite complete section for $c$, a contradiction. ∎

To remedy this, we impose additional requirements on the $E$-coloring as in
the following definition.

######  Definition 3.3.

We call an $E$-coloring $c:[X]^{n}_{E}\to\\{1,\dots,k\\}$ almost transitive if
for any integer $m\geq n$, $A\in[X]^{m}_{E}$ and $B_{1},B_{2}\in[A]^{n-1}$, we
have

$c(\\{z\\}\cup B_{1})=c(\\{z\\}\cup B_{2})$

for all but finitely many $z\in[A]_{E}\setminus(B_{1}\cup B_{2})$, where
$[A]_{E}$ is the $E$-saturation of $A$.

A trivial example of a Borel almost transitive $E$-coloring is the constant
coloring. For an example of a nontrivial Borel almost transitive $E$-coloring,
consider a locally finite Borel graph $G$ on a standard Borel space $X$. Let
$E_{G}$ be the equivalence relation given by the connected components of $G$.
For distinct $x,y\in X$ with $xE_{G}y$, define

$c(\\{x,y\\})=\left\\{\begin{array}[]{ll}1&\mbox{if there is an edge between
$x$ and $y$ in $G$,}\\\ 2&\mbox{otherwise.}\end{array}\right.$

Then $c$ is a Borel almost transitive $E$-coloring of dimension $2$.

We are now ready to prove a Ramsey-type property for hyperfinite nonsmooth
equivalence relations.

######  Theorem 3.4.

Let $X$ be a standard Borel space and let $E$ be an aperiodic hyperfinite
equivalence relation. Let $n,k>1$ be integers. Then for any Borel almost
transitive $E$-coloring $c:[X]^{n}_{E}\to\\{1,\dots,k\\}$, there is a Borel
monochromatic reduction to $E_{1}$ for $c$.

###### Proof.

Let $\\{F_{i}\\}_{i\in\mathbb{N}}$ be an increasing sequence of Borel finite
equivalence relations such that $E=\bigcup_{i}F_{i}$. Without loss of
generality assume $F_{0}$ is the equality relation. Let $C\mapsto<_{C}$ be a
Borel assignment associating to each $E$-class $C$ a linear ordering $<_{C}$
of $C$. Assume that the order type of $(C,<_{C})$ is $\zeta$ for every
$E$-class $C$. We use $\prec$ to denote $\bigcup_{x\in X}(<_{[x]_{E}})$. For
$x\in X$, define

$[x]_{\succ}=\\{y\in[x]_{E}\colon x\prec y\\}.$

Let $c:[X]^{n}_{E}\to\\{1,\dots,k\\}$ be a Borel almost transitive
$E$-coloring. For each $a\in\\{1,\dots,k\\}$, we define an $E$-invariant Borel
set $X_{a}$ and a sequence of partial Borel functions
$\\{f^{a}_{i}\\}_{i\in\mathbb{N}}$, where each $\mbox{\rm dom}(f^{a}_{i})$ is
an $E$-invariant Borel subset of $X$, by induction on $i$ as follows. Let
$f^{a}_{0}(x)=x$ for all $x\in X$ and $a\in\\{1,\dots,k\\}$. For $i\geq 0$,
assume $f^{a}_{0},\dots,f^{a}_{i}$ have been defined so that the following
hold:

  1. (1)

$X=\mbox{\rm dom}(f^{a}_{0})\supseteq\cdots\supseteq\mbox{\rm dom}(f^{a}_{i})$
and each of them is an $E$-invariant Borel subset of $X$;

  2. (2)

for any $x\in\mbox{\rm dom}(f^{a}_{i})$, if we let

$A=\\{f^{a}_{j}(y)\colon y\in[x]_{F_{i}},\ j<i\\},$

then for any $B\in[A]^{n-1}$ with $f^{a}_{i}(x)\not\in B$, we have
$c(\\{f^{a}_{i}(x)\\}\cup B)=a$;

  3. (3)

for any $j\leq i$ and $xF_{j}y\in\mbox{\rm dom}(f^{a}_{i})$, we have
$f^{a}_{j}(x)=f^{a}_{j}(y)$.

Now for any $x\in\mbox{\rm dom}(f^{a}_{i})$, if we let
$K^{a}_{x}=\\{f^{a}_{j}(y)\colon y\in[x]_{F_{i+1}},\ j\leq i\\}$,

$C^{a}_{x}=\\{\,z\in[x]_{E}\colon\mbox{for all $L\in[K^{a}_{x}]^{n-1}$ with
$z\not\in L$, $c(\\{z\\}\cup L)=a$}\,\\},$

and

$A^{a}_{x}=C^{a}_{x}\cap\bigcap\left\\{[y]_{\succ}\colon yF_{i+1}x\right\\},$

then we have two cases according to whether $A^{a}_{x}$ is finite. If
$A^{a}_{x}$ is finite then $f^{a}_{i+1}(x)$ is undefined. Otherwise, define

$f^{a}_{i+1}(x)=\mbox{the $\prec$-least element of }A^{a}_{x}.$

To see that inductive hypothesis (1) is maintained, take $xEy\in\mbox{\rm
dom}(f^{a}_{i})$. By the almost transitivity of $c$ and the finiteness of
$F_{i+1}$, we have that for all but finitely many $z\in[x]_{E}$, $z\in
C^{a}_{x}$ iff $z\in C^{a}_{y}$. Thus $A^{a}_{x}$ is finite iff $A^{a}_{y}$ is
finite, and $x\in\mbox{\rm dom}(f^{a}_{i+1})$ iff $y\in\mbox{\rm
dom}(f^{a}_{i+1})$. It is easier to see that inductive hypotheses (2) and (3)
are also maintained by this construction. This finishes the inductive
definition of $\\{f^{a}_{i}\\}_{i\in\mathbb{N}}$. Then we define
$X_{a}=\bigcap_{i}\mbox{\rm dom}(f^{a}_{i})$. By our construction, if $i\geq
j$ and $xF_{j}y\in X_{a}$, then $f^{a}_{i}(x)=f^{a}_{i}(y)$.

Now, for any $x\in X_{a}$, $i\geq n$ and $I\in[\\{0,\dots,i-1\\}]^{n-1}$, we
have

$c(\\{f_{j}(x)\colon j\in\\{i\\}\cup I\\})=a$

by inductive hypothesis (1). Hence, if we let $S(x)(i)=f^{a}_{i}(x)$ for $x\in
X_{a}$, then $S:X_{a}\to X_{a}^{\mathbb{N}}$ is a Borel monochromatic
reduction to $E_{1}$.

To complete the proof, it suffices to verify that
$X\subseteq\bigcup_{a}X_{a}$. For this, assume $x\not\in\bigcup_{a}X_{a}$.
Then for any $a\in\\{1,\dots,k\\}$, there is $i_{a}$ such that $x\in\mbox{\rm
dom}(f^{a}_{i_{a}})\setminus\mbox{\rm dom}(f^{a}_{i_{a}+1})$. Let

$K=\left\\{f^{a}_{j}(y)\colon a\in\\{1,\dots,k\\},\ j\leq i_{a},\
y\in[x]_{F_{i_{a}+1}}\right\\}.$

Then $K$ is finite. By the almost transitivity of $c$, there is a
$b\in\\{1,\dots,k\\}$ such that for all $L\in[K]^{n-1}$, for all but finitely
many $z\in[x]_{E}\setminus L$, $c(\\{z\\}\cup L)=b$. Thus, at stage $i_{b}$ of
the construction of $X_{b}$, since $x\in\mbox{\rm dom}(f^{b}_{i_{b}})$ and
$K^{b}_{x}\subseteq K$, we have that $A^{b}_{x}$ is infinite. This contradicts
$x\not\in\mbox{\rm dom}(f^{b}_{i_{b}+1})$. ∎

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