On the Chromatic Number of $\mathbb{R}^n$ for Small Values of $n$

The vector graph and the chromatic number of the plane, or how NOT to prove that $χ(\mathbb{E}^2)>4$

The chromatic number $χ\left(\mathcal{E^2}\right)$ of the plane is known to be some integer between 4 and 7, inclusive. We prove a limiting result that says, roughly, that one cannot increase the lower bound on $χ\left(\mathcal{E^2}\right)$ by pasting Moser Spindles together, even countably many.

Box and segment intersection graphs with large girth and chromatic number

We prove that there are intersection graphs of axis-aligned boxes in $\mathbb{R}^3$ and intersection graphs of straight lines in $\mathbb{R}^3$ that have arbitrarily large girth and chromatic number.

Kneser 色函数区分树

The Kneser chromatic function distinguishes trees

本文证明 Kneser 色函数 $X_{K_{\mathbb{N},2}}$ 是树的完全不变量，从而推进了 Stanley 关于色对称函数完全区分树的猜想。

Distance graphs with large chromatic number and arbitrary girth

In this article we consider a problem related to two famous combinatorial topics. One of them concerns the chromatic number of the space. The other deals with graphs having big girth (the length of the shortest cycle) and large chromatic number. Namely, we prove that for any $l\in \mathbb{N}$ there exists a sequence of distance graphs in $\mathbb{R}^n$ with girth at least $l$ and the chromatic number equal to $(c+\bar{o}(1))^n$ with $c>1$.

On the growth rate of chromatic numbers of finite subgraphs

We prove that, for every function $f:\mathbb{N} \rightarrow \mathbb{N}$, there is a graph $G$ with uncountable chromatic number such that, for every $k \in \mathbb{N}$ with $k \geq 3$, every subgraph of $G$ with fewer than $f(k)$ vertices has chromatic number less than $k$. This answers a question of Erdős, Hajnal, and Szemeredi.

具有指数色数和任意围长的几何图

Geometric graphs with exponential chromatic number and arbitrary girth

本文证明了在$\mathbb{R}^d$中存在具有任意大围长且色数至少为$(1.074 + o(1))^d$的单位距离图，改进了已有结果并给出了首个指数底数不随围长趋向1的界限，该结论可推广至其他几何图。

具有密集颜色类的欧几里得空间的色数

The chromatic number of Euclidean space with dense color classes

研究构造欧几里得空间\(\mathbb{R}^n\)的有限着色，使单位距离的点颜色不同且各颜色类稠密，给出\(\mathbb{R}^2\)用12种稠密颜色即可，任意维度\(n\chi(\mathbb{R}^n)+1\)种颜色足够。

The Chromatic Number of $\mathbb{R}^{n}$ with Multiple Forbidden Distances

Let $A\subset\mathbb{R}_{>0}$ be a finite set of distances, and let $G_{A}(\mathbb{R}^{n})$ be the graph with vertex set $\mathbb{R}^{n}$ and edge set $\{(x,y)\in\mathbb{R}^{n}:\ \|x-y\|_{2}\in A\}$, and let $χ(\mathbb{R}^{n},A)=χ\left(G_{A}(\mathbb{R}^{n})\right)$. Erdős asked about the growth rate of the $m$-distance chromatic number \[ \barχ(\mathbb{R}^{n};m)=\max_{|A|=m}χ(\mathbb{R}^{n},A). \] We improve the best existing lower bound for $\barχ(\mathbb{R}^{n};m)$, and show that \[ \barχ(\mathbb{R}^{n};m)\geq\left(Γ_χ\sqrt{m+1}+o(1)\right)^{n} \] where $Γ_χ=0.79983\dots$ is an explicit cons…

\(\mathbb Z^2\) 移位作用的 Schreier 有向图的连续定向色数

The continuous oriented chromatic number of directed Schreier graphs of $\mathbb Z^2$-shift actions

本文证明了 Bernoulli 移位 \(\mathbb Z^2\curvearrowright 2^{\mathbb Z^2}\) 的自由部分上的有向 Schreier 图的连续定向色数为 7。

关于三维切片的色数

On the chromatic numbers of 3-dimensional slices

该研究证明了对任意正ε，三维空间与六维ε区间乘积构成的无限图的色数不小于10，推进了高维切片图色数的下界研究。

阿贝尔群作用的连续边色数

Continuous Edge Chromatic Numbers of Abelian Group Actions

本文研究了阿贝尔群作用的连续边色数，证明了对于$\mathbb{Z}^n$的任意生成集，Bernoulli shift action的Schreier图的连续边色数等于其边色数加1，并给出了标准生成集下的具体数值。

One-point concentration of the clique and chromatic numbers of the random Cayley graph on F_2^n

Green showed that there exist constants $C_1,C_2>0$ such that the clique number $ω$ of the random Cayley graph on $\mathbb{F}_2^n$ satisfies $\lim_{n\to\infty}\mathbb{P}(C_1n\log n < ω< C_2n\log n)=1$. In this paper we find the best possible $C_1$ and $C_2$. Moreover, we prove that for $n$ in a set of density $1$, clique number is actually concentrated on a single value. As a simple consequence of these results, we also prove the one-point concentration result for the chromatic number, thus proving the $\mathbb{F}_2^n$ analogue of the famous conjecture by Bollobás and giving almost the complet…

New Bounds for Chromatic Polynomials and Chromatic Roots

If $G$ is a $k$-chromatic graph of order $n$ then it is known that the chromatic polynomial of $G$, $π(G,x)$, is at most $x(x-1)\cdots (x-(k-1))x^{n-k} = (x)_{\downarrow k}x^{n-k}$ for every $x\in \mathbb{N}$. We improve here this bound by showing that \[ π(G,x) \leq (x)_{\downarrow k} (x-1)^{Δ(G)-k+1} x^{n-1-Δ(G)}\] for every $x\in \mathbb{N},$ where $Δ(G)$ is the maximum degree of $G$. Secondly, we show that if $G$ is a connected $k$-chromatic graph of order $n$ where $k\geq 4$ then $π(G,x)$ is at most $(x)_{\downarrow k}(x-1)^{n-k}$ for every real $x\geq n-2+\left( {n \choose 2} -{k \choose…

Number of vertices in graphs with locally small chromatic number and large chromatic number

We discuss the minimal number of vertices in a graph with a large chromatic number such that each ball of a fixed radius in it has a small chromatic number. It is shown that for every graph $G$ on $\sim((n+rc)/(c+rc))^{r+1}$ vertices such that each ball of radius $r$ is properly $c$-colorable, we have $χ(G)\leq n$.

On the chromatic numbers of small-dimensional Euclidean spaces

The paper is devoted to the study of graph sequence G_n = (V_n, E_n) where V_n is the set of all vectors v in R^n with coordinates from {-1, 0, 1} such that |v| = sqrt(3), and E_n consists of all pairs of vertices with the scalar product 1. We find exactly the independence number of G_n. As a corollary we get some new lower bounds of chi(\R^n) and chi(\Q^n) for small values of n.

An equivalent formulation of chromatic quasi-polynomials

Given a central integral arrangement, the reduction of the arrangement modulo positive integers $q$ gives rise to a subgroup arrangement in $(\mathbb{Z}/q\mathbb{Z})^\ell$. Kamiya-Takemura-Terao (2008) introduced the notion of characteristic quasi-polynomials, which uses to evaluate the cardinality of the complement of the subgroup arrangement. Chen-Wang (2012) found a similar but more general setting that replacing the integral arrangement by its restriction to a subspace of $\mathbb{R}^\ell$, and evaluating the cardinality of the $q$-reduction complement will also lead to a quasi-polynomial …

关于热带范数下 $\mathbb{R}^n$ 的色数和等距维数

On the chromatic number and equilateral dimension of $\mathbb{R}^n$ with the tropical norm

研究热带范数下Rn的色数（Hadwiger-Nelson问题的热带类比）和等距维数，给出上界2^n，渐近界Θ(2^n/√n)，证明χ_tr(R^3)=8，并猜想等距维数等于Sperner反链界。

Expected Chromatic Number of Random Subgraphs

Given a graph $G$ and $p \in [0,1]$, let $G_p$ denote the random subgraph of $G$ obtained by keeping each edge independently with probability $p$. Alon, Krivelevich, and Sudokov proved $\mathbb{E} [χ(G_p)] \geq C_p \frac{χ(G)}{\log |V(G)|}$, and Bukh conjectured an improvement of $\mathbb{E}[χ(G_p)] \geq C_p \frac{χ(G)}{\log χ(G)}$. We prove a new spectral lower bound on $\mathbb{E}[χ(G_p)]$, as progress towards Bukh's conjecture. We also propose the stronger conjecture that for any fixed $p \leq 1/2$, among all graphs of fixed chromatic number, $\mathbb{E}[χ(G_p)]$ is minimized by the complet…

$S$-packing colorings of distance graphs $G(\mathbb{Z},\{2,t\})$

Given a graph $G$ and a non-decreasing sequence $S=(a_1,a_2,\ldots)$ of positive integers, the mapping $f:V(G) \rightarrow \{1,\ldots,k\}$ is an $S$-packing $k$-coloring of $G$ if for any distinct vertices $u,v\in V(G)$ with $f(u)=f(v)=i$ the distance between $u$ and $v$ in $G$ is greater than $a_i$. The smallest $k$ such that $G$ has an $S$-packing $k$-coloring is the $S$-packing chromatic number, $χ_S(G)$, of $G$. In this paper, we consider the distance graphs $G(\mathbb{Z},\{2,t\})$, where $t>1$ is an odd integer, which has $\mathbb{Z}$ as its vertex set, and $i,j\in\mathbb{Z}$ are adjacent…

线性方程和递归的色数阈值

Chromatic thresholds for linear equations and recurrence

研究有限域上线性方程解自由集的色数阈值，确定其为零的充要条件，并解决Griesmer问题。

Almost-monochromatic sets and the chromatic number of the plane

In a colouring of $\mathbb{R}^d$ a pair $(S,s_0)$ with $S\subseteq \mathbb{R}^d$ and with $s_0\in S$ is \emph{almost monochromatic} if $S\setminus \{s_0\}$ is monochromatic but $S$ is not. We consider questions about finding almost monochromatic similar copies of pairs $(S,s_0)$ in colourings of $\mathbb{R}^d$, $\mathbb{Z}^d$, and in $\mathbb{Q}$ under some restrictions on the colouring. Among other results, we characterise those $(S,s_0)$ with $S\subseteq \mathbb{Z}$ for which every finite colouring of $\mathbb{R}$ without an infinite monochromatic arithmetic progression contains an almost mo…

Subgraphs of large connectivity and chromatic number

Resolving a problem raised by Norin, we show that for each $k \in \mathbb{N}$, there exists an $f(k) \le 7k$ such that every graph $G$ with chromatic number at least $f(k)+1$ contains a subgraph $H$ with both connectivity and chromatic number at least $k$. This result is best-possible up to multiplicative constants, and sharpens earlier results of Alon-Kleitman-Thomassen-Saks-Seymour from 1987 showing that $f(k) = O(k^3)$, and of Chudnovsky-Penev-Scott-Trotignon from 2013 showing that $f(k) = O(k^2)$. Our methods are robust enough to handle list colouring as well: we also show that for each $k…

关于染色Delaunay镶嵌的规模

On the Size of Chromatic Delaunay Mosaics

该研究提出了高维染色Delaunay镶嵌概念，在d、s为常数的假设下推导了其规模界，给出随机染色、Delone集、Poisson点过程及二维稠密集等不同情形下的胞腔数量级结果，为其应用提供理论支撑。

New construction of graphs with high chromatic number and small clique

In this note, we introduce a new method for constructing graphs with high chromatic number and small clique. Indeed, via this method, we present a new proof for the well-known Kneser's conjecture.

Regions without complex zeros for chromatic polynomials on graphs with bounded degree

We prove that the chromatic polynomial $P_\mathbb{G}(q)$ of a finite graph $\mathbb{G}$ of maximal degree $\D$ is free of zeros for $\card q\ge C^*(\D)$ with $$ C^*(\D) = \min_{0<x<2^{1\over \D}-1} {(1+x)^{\D-1}\over x [2-(1+x)^\D]} $$ This improves results by Sokal (2001) and Borgs (2005). Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.

On graphs with a large chromatic number containing no small odd cycles

In this paper, we present the lower bounds for the number of vertices in a graph with a large chromatic number containing no small odd cycles.

枚举色可选性

Enumerative Chromatic Choosability

本文研究枚举色可选性，完全刻画了色数为二的枚举色可选图，构造了色可选但非枚举色可选的例子，并探讨了任意图与完全图连接后成为枚举色可选图的猜想。

A small 6-chromatic two-distance graph in the plane

We give a new, simple proof for the lower bound of the chromatic number of the Euclidean plane with two forbidden distances, based on a graph with only 16 vertices.

On the chromatic number of the preferential attachment graph

We prove that for every $m\in \mathbb N$ and every $δ\in (-m,0)$, the chromatic number of the preferential attachment graph $PA_t(m, δ)$ is asymptotically almost surely equal to $m+1$. The proof relies on a combinatorial construction of a family of digraphs of chromatic number $m+1$ followed by a proof that asymptotically almost surely there is a digraph in this family, which is realised as a subgraph of the preferential attachment graph.

色对称拟对称函数的一种幂零实现

A unipotent realization of the chromatic quasisymmetric function

本文通过有限一般线性群 $\mathrm{GL}_{n}(\mathbb{F}_{q})$ 的复特征标理论实现了色对称拟对称函数和垂直条带 LLT 多项式，提出了一种基于幂零上三角群归纳和 Hopf 代数的计算方法，并揭示了其与 Hessenberg 簇的联系。

平面的分数色数至少为4

The fractional chromatic number of the plane is at least 4

本文证明了欧几里得平面的分数色数至少为4，通过引入几何分数色数并构造27顶点的单位距离图实现，同时证明了有限分数色数与霍尔比相等，并猜想独立比1/4不可达。

On Packing Colorings of Distance Graphs

The {\em packing chromatic number} $χ_ρ(G)$ of a graph $G$ is the least integer $k$ for which there exists a mapping $f$ from $V(G)$ to $\{1,2,\ldots ,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. This paper studies the packing chromatic number of infinite distance graphs $G(\mathbb{Z},D)$, i.e. graphs with the set $\mathbb{Z}$ of integers as vertex set, with two distinct vertices $i,j\in \mathbb{Z}$ being adjacent if and only if $|i-j|\in D$. We present lower and upper bounds for $χ_ρ(G(\mathbb{Z},D))$, showing that for finite $D$, the packing chromatic number i…

Chromatic Bounds On Orbital Chromatic Roots

Given a group $G$ of automorphisms of a graph $Γ$, the orbital chromatic polynomial $OP_{Γ,G}(x)$ is the polynomial whose value at a positive integer $k$ is the number of orbits of $G$ on proper $k$-colorings of $Γ.$ In \cite{Cameron}, Cameron et. al. explore the roots of orbital chromatic polynomials, and in particular prove that orbital chromatic roots are dense in $\mathbb{R}$, extending Thomassen's famous result (see \cite{Thomassen}) that chromatic roots are dense in $[\frac{32}{27},\infty)$. Cameron et al \cite{Cameron} further conjectured that the real roots of the orbital chromatic pol…

The list chromatic number of graphs with small clique number

We prove that every triangle-free graph with maximum degree $Δ$ has list chromatic number at most $(1+o(1))\fracΔ{\ln Δ}$. This matches the best-known bound for graphs of girth at least 5. We also provide a new proof that for any $r\geq 4$ every $K_r$-free graph has list-chromatic number at most $200r\frac{Δ\ln\lnΔ}{\lnΔ}$.

Examples of topologically highly chromatic graphs with locally small chromatic number

Kierstead, Szemerédi, and Trotter showed that a graph with at most $\lfloor r/(2n)\rfloor^n$ vertices such that each ball of radius $r$ in it is $c$-colorable should have chromatic number at most $n(c-1)+1$. We show that this estimate is sharp in $r$. Namely, for every $n$, $r$, and $c$ we construct a graph $G$ containing $O((2rc)^{n-1}c)$ vertices such that $χ(G)\geq n(c-1)+1$, although each ball of radius $r$ in $G$ is $c$-colorable. The core idea is the construction of a graph whose neighborhood complex is homotopy equivalent to the join of neighborhood complexes of two given graphs.

θ图的锥是枚举色可选的

Cones of Theta Graphs are Enumeratively Chromatic-Choosable

本文证明当G为θ图时，K₁∨G是枚举色可选的，其证明结合经典不等式、多项式方法与DP着色思想，回应了枚举色可选性的类似Ohba结论的猜想。

The intersection graph of ideals of $\mathbb{Z}_n$ is\\ weakly perfect

A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let $R$ be a ring and $I(R)^*$ be the set of all left proper non-trivial ideals of $R$. The intersection graph of ideals of $R$, denoted by $G(R)$, is a graph with the vertex set $I(R)^*$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J\neq 0$. In this paper, it is shown that $G(\mathbb{Z}_n)$, for every positive integer $n$, is a weakly perfect graph. Also, for some values of $n$, we give an explicit formula for the vertex chromatic number of $G(\mathbb{Z}_n)$. Furthermore, it i…

The Chromatic Number of Ordered Graphs With Constrained Conflict Graphs

An ordered graph $G$ is a graph whose vertex set is a subset of integers. The edges are interpreted as tuples $(u,v)$ with $u < v$. For a positive integer $s$, a matrix $M \in \mathbb{Z}^{s \times 4}$, and a vector $\mathbf{p} = (p,\ldots,p) \in \mathbb{Z}^s$ we build a conflict graph by saying that edges $(u,v)$ and $(x,y)$ are conflicting if $M(u,v,x,y)^\top \geq \mathbf{p}$ or $M(x,y,u,v)^\top \geq \mathbf{p}$, where the comparison is componentwise. This new framework generalizes many natural concepts of ordered and unordered graphs, such as the page-number, queue-number, band-width, interv…

Deformations of box complexes

Box complex is a $\mathbb{Z}_2$-space associated to a graph, and it is known that a certain $\mathbb{Z}_2$-homotopy invariant of it, called the $\mathbb{Z}_2$-index, gives an effective lower bound for the chromatic number. On the other hand, we show that any $\mathbb{Z}_2$-homotopy invariant of the box complex is not equivalent to the chromatic number. Namely, we construct a graph homomorphism $f:X \rightarrow Y$ such that it gives rise to a $\mathbb{Z}_2$-homotopy equivalence between their box complexes, but $X$ and $Y$ have different chromatic numbers. To see this, we show that some deformat…

The Chromatic Number of the $q$-Kneser Graph for $q \geq 5$

We obtain a new weak Hilton-Milner type result for intersecting families of $k$-spaces in $\mathbb{F}_q^{2k}$, which improves several known results. In particular the chromatic number of the $q$-Kneser graph $qK_{n:k}$ was previously known for $n > 2k$ (except for $n=2k+1$ and $q=2$) or $k < q \log q - q$. Our result determines the chromatic number of $qK_{2k:k}$ for $q \geq 5$, so that the only remaining open cases are $(n, k) = (2k, k)$ with $q \in \{ 2, 3, 4 \}$ and $(n, k) = (2k+1, k)$ with $q = 2$.

Burling graphs, chromatic number, and orthogonal tree-decompositions

A classic result of Asplund and Grünbaum states that intersection graphs of axis-aligned rectangles in the plane are $χ$-bounded. This theorem can be equivalently stated in terms of path-decompositions as follows: There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that every graph that has two path-decompositions such that each bag of the first decomposition intersects each bag of the second in at most $k$ vertices has chromatic number at most $f(k)$. Recently, Dujmović, Joret, Morin, Norin, and Wood asked whether this remains true more generally for two tree-decompositions. In this note…

On the cromatic number of infinitesimal plane layer

We consider natural generalization of plane chromatic number problem. We consider chromatic numbers $χ$ of spaces $\mathbb{R}^n \times [0,\varepsilon]^k$ for arbitrary small $\varepsilon$. We prove that $5 \leqχ(\mathbb{R}^2\times [0,\varepsilon])\leq 7$ and ${6\leq χ(\mathbb{R}^2\times [0,\varepsilon]^2) \leq 7}$. Also we consider natural questions, arising from this considerations.

Distance graphs having large chromatic numbers and not containing cliques or cycles of given size

In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in $ {\mathbb R}^n $, i.e., graphs such that their vertices are vectors and their edges are pairs of vectors at a distance from a given set of postive numbers apart. A series of new lower bounds are obtained for the chromatic numbers of such graphs with different restrictions on the clique numbers and the girths.

Computing Small Unit-Distance Graphs with Chromatic Number 5

We present a new method for reducing the size of graphs with a given property. Our method, which is based on clausal proof minimization, allowed us to compute several 553-vertex unit-distance graphs with chromatic number 5, while the smallest published unit-distance graph with chromatic number 5 has 1581 vertices. The latter graph was constructed by Aubrey de Grey to show that the chromatic number of the plane is at least 5. The lack of a 4-coloring of our graphs is due to a clear pattern enforced on some vertices. Also, our graphs can be mechanically validated in a second, which suggests that…

The Distant-l Chromatic Number of Random Geometric Graphs

A random geometric graph $G_n$ is given by picking $n$ vertices in $\mathbb{R}^d$ independently under a common bounded probability distribution, with two vertices adjacent if and only if their $l^p$-distance is at most $r_n$. We investigate the distant-$l$ chromatic number $χ_l(G_n)$ of $G_n$ for $l\ge1$. Complete picture of the ratios of $χ_l(G_n)$ to the chromatic number $χ(G_n)$ are given in the sense of almost sure convergence.

小图的环色数

Circular chromatic index of small graphs

本研究系统确定了最大度数为4、5、6的小图和多重图的环色数，并推翻了关于环色数接近Δ+1的猜想。

不可测着色的可测障碍

Measurable obstructions for unmeasurable colourings

针对Székely提出的欧氏空间可测与普通色数是否不同的问题，该研究通过建立相关公式，将可测色数的诸多结果提升至普通色数，并改进了相关下界，还拓展了相关定理的适用范围。

Even factors in edge-chromatic-critical graphs with a small number of divalent vertices

A finite simple connected graph $G$ with maximum degree $k$ is $k$-critical if it has chromatic index $χ'(G)=k+1$ and $χ'(G-e)=k$ for every edge $e\in E(G)$. Bej and the first author raised the question whether every $k$-critical graph has an even factor. We prove that every $k$-critical graph with at most $2k-6$ vertices of degree 2 has an even factor.

典型范数的Hadwiger--Nelson问题

Hadwiger--Nelson Problem for Typical Norms

本文研究典型范数的Hadwiger--Nelson问题，将单位距离图色数的上界从指数改进为线性（至多2d且紧），并借助高维矩阵推广的Lonely runner猜想解决了相关猜想。

Total dominator chromatic number of $k$-subdivision of graphs

Let $G$ be a simple graph. A total dominator coloring of $G$, is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number $χ_d^t(G)$ of $G$, is the minimum number of colors among all total dominator coloring of $G$. For any $k \in \mathbb{N}$, the $k$-subdivision of $G$ is a simple graph $G^{\frac{1}{k}}$ which is constructed by replacing each edge of $G$ with a path of length $k$. In this paper, we study the total dominator chromatic number of $k$-subdivision of $G$.

汉明图及其推广的最小特征值和量子色数

On the Smallest Eigenvalues and Quantum Chromatic Numbers of Hamming Graphs and Generalizations

本文研究了距离参数小于一半长度的汉明图的最小特征值，给出了渐近下界，并应用于量子色数的下界。

Colorful Subhypergraphs in Uniform Hypergraphs

There are several topological results ensuring the existence of a large complete bipartite subgraph in any properly colored graph satisfying some special topological regularity conditions. In view of $\mathbb{Z}_p$-Tucker lemma, Alishahi and Hajiabolhassan [{\it On the chromatic number of general Kneser hypergraphs, Journal of Combinatorial Theory, Series B, 2015}] introduced a lower bound for the chromatic number of Kneser hypergraphs ${\rm KG}^r({\mathcal H})$. Next, Meunier [{\it Colorful subhypergraphs in Kneser hypergraphs, The Electronic Journal of Combinatorics, 2014}] improved their re…

射影空间的正常四边形的着色

Colouring normal quadrangulations of projective spaces

本文证明对d=3存在色数任意大的射影空间正常四边形剖分，并证明任意维度下色数不可能为3，否定了几何条件可去除的猜想。

4维格的色数

The chromatic number of 4-dimensional lattices

本研究利用4维52种平行多面体分类，结合格Cayley图同构检验、诱导子图下界、周期着色上界及SAT求解器，确定了所有4维格的色数。