arXiv subjects

Edward Hou

Publications and source records attributed to Edward Hou.

4 recordsLinked to original sources

arXiv2026

Randomized Borel $(2d+1)$-coloring of digraphs

Let $G$ be a Borel digraph with maximum out-degree $d \in \mathbb{N}$. We show that $G$ admits a random Borel $(2d+1)$-coloring for which every edge is almost surely not monochromatic. This gives a simpler proof of a recent result of Pelayo-Gómez: such a graph $G$ admits a measurable proper $(2d+1)$-coloring with respect to any Borel probability measure on $V(G)$. Our proof is an adaptation of Pelayo-Gómez's proof to the randomized Borel setting.

math.LO↗

arXiv2025

Measurable domatic partitions

Let $Γ$ be a compact Polish group of finite topological dimension. For a countably infinite subset $S\subseteq Γ$, a domatic $\aleph_0$-partition (for its Schreier graph on $Γ$) is a partial function $f:Γ\rightharpoonup\mathbb{N}$ such that for every $x\in Γ$, one has $f[S\cdot x]=\mathbb{N}$. We show that a continuous domatic $\aleph_0$-partition exists, if and only if a Baire measurable domatic $\aleph_0$-partition exists, if and only if the topological closure of $S$ is uncountable. A Haar measurable domatic $\aleph_0$-partition exists for all choices of $S$. We also investigate domatic partitions in the general descriptive graph combinatorial setting.

math.LO↗

arXiv2025

A note on measure-theoretic domatic partitions

We show that if $(X,μ)$ is a standard probability space, then every $μ$-preserving $\aleph_0$-regular Borel graph on $X$ admits a $μ$-measurable vertex $\aleph_0$-coloring in which every vertex sees every color in its neighborhood.

math.LO↗

arXiv2025

Borel Polychromatic Number of Grids

We study Borel polychromatic colorings of grid graphs arising from free Borel actions of $\mathbb{Z}^d$. A polychromatic coloring is one in which every unit $d$-dimensional cube sees all available colors. In the classical setting, every grid admits a $2^d$-polychromatic coloring, while in the Borel setting this fails. Our main result shows that every free $\mathbb{Z}^d$-action admits a Borel $(2^d-1)$-polychromatic coloring. This result is sharp: any action where the generators act ergodically does not admit a Borel $2^d$-polychromatic coloring. We conclude with open directions for extending the theory beyond cube tilings and for exploring the dependence of Borel polychromatic numbers on the underlying action.

math.LO↗