Related papers: Optimization Tools for Computing Colorings of $[1,…

We study the number of monochromatic solutions to linear equations in a $2$-coloring of $\{1,\ldots,n\}$. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any $2$-coloring of…

Let $R_k(H;K_m)$ be the smallest number $N$ such that every coloring of the edges of $K_{N}$ with $k+1$ colors has either a monochromatic $H$ in color $i$ for some $1\leqslant i\leqslant k$, or a monochromatic $K_{m}$ in color $k+1$. In…

We show that, for every $r, k$, there is an $n = n(r,k)$ so that any $r$-coloring of the edges of the complete graph on $[n]$ will yield a monochromatic complete subgraph on vertices ${a + \sum_{i \in I} d_i \mid I \subseteq [k]}$ for some…

One formulation of the Erdos-Szekeres monotone subsequence theorem states that for any red/blue coloring of the edge set of the complete graph on $\{1, 2, \ldots, N\}$, there exists a monochromatic red $s$-clique or a monochromatic blue…

For positive integers $a_1,a_2,...,a_m$, we determine the least positive integer $R(a_1,...,a_m)$ such that for every 2-coloring of the set $[1,n]={1,...,n}$ with $n\ge R(a_1,...,a_m)$ there exists a monochromatic solution to the equation…

Given a graph $H$ and a positive integer $k$, the {\it $k$-colored Ramsey number} $R_k(H)$ is the minimum integer $n$ such that in every $k$-edge-coloring of the complete graph $K_{n}$, there is a monochromatic copy of $H$. Given two graphs…

Let $K\_{[k,t]}$ be the complete graph on $k$ vertices from which a set of edges, induced by a clique of order $t$, has been dropped. In this note we give two explicit upper bounds for $R(K\_{[k\_1,t\_1]},\dots, K\_{[k\_r,t\_r]})$ (the…

For a set of positive integers $A \subseteq [n]$, an $r$-coloring of $A$ is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of sum-free…

The size-Ramsey number $\hat{R}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In…

For any linear inequality in three variables $\mathcal{L}$, we determine (if it exist) the smallest integer $R(\mathcal{L}, \mathbb{Z}/3\mathbb{Z})$ such that: for every mapping $\chi :[1,n] \to \{0,1,2\}$, with $n\geq R(\mathcal{L},…

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…

In 1955, Greenwood and Gleason showed that the Ramsey number R(3, 3, 3) = 17 by constructing an edge-chromatic graph on 16 vertices in three colors with no triangles. Their technique employed finite fields. This same result was obtained…

We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on $n$ vertices with min O(Delta^{1/3} log^{1/2} Delta log n),…

In this paper, we introduce Euclidean Gallai-Ramsey theory, by combining Euclidean Ramsey theory and Gallai-Ramsey theory on graphs. More precisely, we consider the following problem: For an integer $r$ and configurations $K$ and $K'$, does…

We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of $[1,n]$, of monochromatic $\{x,y,x+ay\}$ triples for $a \geq 1$. We give a new proof of the original case of $a=1$. We show that the minimum…

Let the integers $1,\ldots,n$ be assigned colors. Szemer\'edi's theorem implies that if there is a dense color class then there is an arithmetic progression of length three in that color. We study the conditions on the color classes forcing…

We provide a general framework to construct colorings avoiding short monochromatic arithmetic progressions in Euclidean Ramsey theory. Specifically, if $\ell_m$ denotes $m$ collinear points with consecutive points of distance one apart, we…

For any countably infinite graph $G$, Ramsey's theorem guarantees an infinite monochromatic copy of $G$ in any $r$-coloring of the edges of the countably infinite complete graph $K_\mathbb{N}$. Taking this a step further, it is natural to…

We study the mixed Ramsey number maxR(n,K_m,K_r), defined as the maximum number of colours in an edge-colouring of the complete graph K_n, such that K_n has no monochromatic complete subgraph on m vertices and no rainbow complete subgraph…

Let $R_h(k; \ell)$ be the smallest integer $n$ such that any edge coloring of a complete graph on $n$ vertices in $\ell$ colors results in a monochromatic $K_k$-minor, in other words, a graph with Hadwiger number $k$, i.e., a graph that…