Related papers: On colorings of variable words

We prove that with high probability over the choice of a random graph $G$ from the Erd\H{o}s-R\'enyi distribution $G(n,1/2)$, a natural $n^{O(\varepsilon^2 \log n)}$-time, degree $O(\varepsilon^2 \log n)$ sum-of-squares semidefinite program…

We present a ``reasonable'' statement about Lie algebras that is equivalent to the Four Color Theorem. The notions appearing in the statement also appear in the theory of finite-type invariants of knots (Vassiliev invariants) and…

Consider edge colorings of digraphs where edges $v_1 v_2$ and $v_2 v_3$ have different colors. This coloring induces a vertex coloring by sets of edge colors, in which edge $v_1 v_2$ in the graph implies that the set color of $v_1$ contains…

Consider the graph $\mathbb{H}(d)$ whose vertex set is the hyperbolic plane, where two points are connected with an edge when their distance is equal to some $d>0$. Asking for the chromatic number of this graph is the hyperbolic analogue to…

The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are…

Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…

In this note we study graphs $G_r$ with the property that every colouring of $E(G_r)$ with $r+1$ colours admits a copy of some graph $H$ using at most $r$ colours. For $1\le r\le e(H)$ such graphs occur naturally at intermediate steps in…

Following D.B. Karaguezian, V. Reiner, and M.L. Wachs (Matching Complexes, Bounded Degree Graph Complexes, and Weight Spaces of $GL$-Complexes, Journal of Algebra 2001) we study the connectivity degree and shellability of multiple…

We prove that the statement "for every infinite cardinal nu, every graph with list chromatic nu has coloring number at most beth_omega (nu)" proved by Kojman [6] using the RGCH theorem [11] implies the RGCG theorem via a short forcing…

We use a graph-theoretic approach which yields improvements on the known Gilbert-Varshamov (GV) bound for sum-rank-metric codes for certain parameters. In particular, we show that asymptotically $\mathbb{F}_q^{\mathbf{n} \times \mathbf{m}}$…

In 2009, Blagojevic, Matschke & Ziegler established the first tight colored Tverberg theorem, but no lower bounds for the number of colored Tverberg partitions. We develop a colored version of our previous results (2008), and we extend our…

We show that for any $i > 0$, it is decidable, given a regular language, whether it is expressible in the $\Sigma_i[<]$ fragment of first-order logic FO[<]. This settles a question open since 1971. Our main technical result relies on the…

For a graph $G$ with a list assignment $L$ and two $L$-colorings $\alpha$ and $\beta$, an $L$-recoloring sequence from $\alpha$ to $\beta$ is a sequence of proper $L$-colorings where consecutive colorings differ at exactly one vertex. We…

The degree chromatic polynomial $Pm(G,k)$ of a graph $G$ counts the number of $k$-colorings in which no vertex has $m$ adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree…

The first-order part of the Ramsey's Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey's Theorem for pairs into the weaker principles, EM (the…

An ordered graph $G$ is a graph together with a specified linear ordering on the vertices, and its interval chromatic number is the minimum number of independent sets consisting of consecutive vertices that are needed to partition the…

An edge-coloring of a graph $G$ with colors $1,...,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval…

P. Kirchberger proved that, for a finite subset $X$ of $\mathbb{R}^{d}$ such that each point in $X$ is painted with one of two colors, if every $d+2$ or fewer points in $X$ can be separated along the colors, then all the points in $X$ can…

We extend two well-known results in Ramsey theory from from $K_n$ to arbitrary $n$-chromatic graphs. The first is a note of Erd\H os and Rado stating that in every 2-coloring of the edges of $K_n$ there is a monochromatic tree on $n$…

We consider a language together with the subword relation, the cover relation, and regular predicates. For such structures, we consider the extension of first-order logic by threshold- and modulo-counting quantifiers. Depending on the…