Brooks's theorem for measurable colorings

Measurable versions of the Lovász Local Lemma and measurable graph colorings

In this paper we investigate the extent to which the Lovász Local Lemma (an important tool in probabilistic combinatorics) can be adapted for the measurable setting. In most applications, the Lovász Local Lemma is used to produce a function $f \colon X \to Y$ with certain properties, where $X$ is some underlying combinatorial structure and $Y$ is a (typically finite) set. Can this function $f$ be chosen to be Borel or $μ$-measurable for some probability Borel measure $μ$ on $X$ (assuming that $X$ is a standard Borel space)? In the positive direction, we prove that if the set of constraints put…

Equitable Colorings of Borel Graphs

Hajnal and Szemerédi proved that if $G$ is a finite graph with maximum degree $Δ$, then for every integer $k \geqslant Δ+1$, $G$ has a proper coloring with $k$ colors in which every two color classes differ in size at most by $1$; such colorings are called equitable. We obtain an analog of this result for infinite graphs in the Borel setting. Specifically, we show that if $G$ is an aperiodic Borel graph of finite maximum degree $Δ$, then for each $k \geqslant Δ+ 1$, $G$ has a Borel proper $k$-coloring in which every two color classes are related by an element of the Borel full semigroup of $G$…

可测正则子图

Measurable Regular Subgraphs

该研究刻画了d-正则二部Borel图存在Baire可测k-正则生成子图的充要条件，给出首个Borel图上有Baire可测解但高度可计算图上无可计算解的局部可检验着色问题，并证明超有限图测度情形下的类似结论。

Borel fractional colorings of Schreier graphs

Let $Γ$ be a countable group and let $G$ be the Schreier graph of the free part of the Bernoulli shift of $Γ$ (with respect to some finite subset $F \subseteq Γ$). We show that the Borel fractional chromatic number of $G$ is equal to $1$ over the measurable independence number of $G$. As a consequence, we asymptotically determine the Borel fractional chromatic number of $G$ when $Γ$ is the free group, answering a question of Meehan.

FIID 染色随机映射

FIID Coloring Random Maps

本文证明方格渗流连通分量可被 FIID 因子 5-染色，临界概率之上可 4-染色，并推广了相关对应关系及平面性定义。

Approximate Schreier decorations and approximate Kőnig's line coloring Theorem

Following recent result of L. M. T\' oth [arXiv:1906.03137] we show that every $2Δ$-regular Borel graph $\mathcal{G}$ with a (not necessarily invariant) Borel probability measure admits approximate Schreier decoration. In fact, we show that both ingredients from the analogous statements for finite graphs have approximate counterparts in the measurable setting, i.e., approximate Kőnig's line coloring Theorem for Borel graphs without odd cycles and approximate balanced orientation for even degree Borel graphs.

有向图的随机 Borel $(2d+1)$-着色

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

针对最大出度为 $d$ 的 Borel 有向图，证明存在随机 Borel $(2d+1)$-着色使每条边几乎必然非单色，并由此简化了可测正常着色的已知结果。

从描述性到分布式

From descriptive to distributed

本综述从描述集合论视角出发，聚焦图的顶点与边着色问题，介绍该领域成果向分布式计算的转化、相关技术发展及开放问题，还提出了次指数增长图上布鲁克斯着色的高效确定性分布式算法。

可测的Vizing定理

Measurable Vizing's theorem

本文证明了有界度Borel图的完整可测版Vizing定理，解决了Marks及Kechris和Marks提出的问题，并扩展了此前仅在测度不变假设下的结果。

广义康托尔空间中$G_0$-二分法的失败

Failure of the $G_0$-dichotomy for generalized Cantor spaces

本文证明广义康托尔空间中$G_0$-二分法的类比在闭图这一最低复杂度层次上不成立。

不平衡图中的可测匹配

Measurable matchings in unbalanced graphs

针对局部有限二分不平衡图，利用描述集合论和概率方法，证明了存在Borel匹配覆盖几乎所有A中顶点，并推广到完全覆盖及边着色等应用。

$2n+1$ 问题的测度一半

The measure half of the $2n+1$ problem

本文证明任意 Borel 函数生成的图在测度意义下具有最优的 $2n+1$ 着色，肯定回答了 Kechris 等提出的测度版本问题，且不依赖不变性或局部有限性。