Related papers: Galvin's problem in higher dimensions

In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a…

We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel's circle problem (1959). The proof relies on a…

We prove the existence of a function $f :\mathbb{N} \to \mathbb{N}$ such that the vertices of every planar graph with maximum degree $\Delta$ can be 3-colored in such a way that each monochromatic component has at most $f(\Delta)$ vertices.…

We study a generalisation of Vizing's theorem, where the goal is to simultaneously colour the edges of graphs $G_1,\dots,G_k$ with few colours. We obtain asymptotically optimal bounds for the required number of colours in terms of the…

Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points of distance one. For every natural number $n$, we prove the existence of a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of…

It is consistent that for every monotonically increasing function f:omega->omega there is a graph with size and chromatic number aleph_1 in which every n-chromatic subgraph has at least f(n) elements (n >= 3). This solves a $250 problem of…

We study several basic problems about colouring the $p$-random subgraph $G_p$ of an arbitrary graph $G$, focusing primarily on the chromatic number and colouring number of $G_p$. In particular, we show that there exist infinitely many…

An edge coloring of a graph $G$ is a Gallai coloring if it contains no rainbow triangle. We show that the number of Gallai $r$-colorings of $K_n$ is $\left(\binom{r}{2}+o(1)\right)2^{\binom{n}{2}}$. This result indicates that almost all…

Gallai's colouring theorem states that if the edges of a complete graph are 3-coloured, with each colour class forming a connected (spanning) subgraph, then there is a triangle that has all 3 colours. What happens for more colours: if we…

We provide a simplified proof of the following special case of Wegner's conjecture: every planar graph of maximum degree at most three admits a distance-2 coloring with at most eight colors. Our main contribution is significant…

We examine maximum vertex coloring of random geometric graphs, in an arbitrary but fixed dimension, with a constant number of colors. Since this problem is neither scale-invariant nor smooth, the usual methodology to obtain limit laws…

Let $\Gamma$ be an Abelian group and let $G$ be a simple graph. We say that $G$ is $\Gamma$-colorable if for some fixed orientation of $G$ and every edge labeling $\ell:E(G)\rightarrow \Gamma$, there exists a vertex coloring $c$ by the…

We prove that for every integer $k$, every finite set of points in the plane can be $k$-colored so that every half-plane that contains at least $2k-1$ points, also contains at least one point from every color class. We also show that the…

Let $G$ be a graph and c a proper k-coloring of G, i.e. any two adjacent vertices u and v have different colors c(u) and c(v). A proper k-coloring is a b-coloring if there exists a vertex in every color class that contains all the colors in…

Total coloring of a graph is a coloring of its vertices and edges such that adjacent or incident elements receive distinct colors. Total coloring conjecture (stipulating that the total chromatic number of a graph $G$ is at most…

We compute the Hadwiger-Nelson numbers $\chi(E^2)$ for certain number fields $E$, that is, the smallest number of colors required to color the points in the plane with coordinates in~$E$ so that no two points at distance $1$ from one…

Suppose that $\mathbb{N}$ is $2$-coloured. Then there are infinitely many monochromatic solutions to $x + y = z^2$. On the other hand, there is a $3$-colouring of $\mathbb{N}$ with only finitely many monochromatic solutions to this…

Let $X$ be a compact metric space and let $|A|$ denote the cardinality of a set $A$. We prove that if $f\colon X\to X$ is a homeomorphism and $|X|=\infty$ then for all $\delta>0$ there is $A\subset X$ such that $|A|=4$ and for all $k\in Z$…

We prove that for all $\varepsilon>0$, there exists a positive integer $n_0$ such that if $G$ is a graph on $n\geq n_0$ vertices with $\delta(G)\geq\tfrac{1}{2}(1 + \varepsilon)n$, then $G$ satisfies the Total Coloring Conjecture, that is,…

We give asymptotically optimal constructions in generalized Ramsey theory using results about conflict-free hypergraph matchings. For example, we present an edge-coloring of $K_{n,n}$ with $2n/3 + o(n)$ colors such that each $4$-cycle…