Related papers: A note on multidimensional Ramsey numbers

The multicolor Ramsey number $r_k(F)$ of a graph $F$ is the least integer $n$ such that in every coloring of the edges of $K_n$ by $k$ colors there is a monochromatic copy of $F$. In this short note we prove an upper bound on $r_k(F)$ for a…

The generalized Ramsey number $r(G, H, q)$ is the minimum number of colors needed to color the edges of $G$ such that every isomorphic copy of $H$ has at least $q$ colors. In this note, we improve the upper and lower bounds on $r(K_{n, n},…

The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for $2\leq k\in \mathbb{N}$, any colouring of the edges of $K_n$ with $n$ sufficiently large gives a copy of $C_{2k}$ which has one of three canonical colour…

Let H_1, ..., H_k be graphs. The multicolor Ramsey number r(H_1,...,H_k) is the minimum integer r such that in every edge-coloring of K_r by k colors, there is a monochromatic copy of H_i in color i for some 1 <= i <= k. In this paper, we…

The canonical Ramsey theorem of Erd\H{o}s and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy…

We determine the Ramsey number of a connected clique matching. That is, we show that if $G$ is a $2$-edge-coloured complete graph on $(r^2 - r - 1)n - r + 1$ vertices, then there is a monochromatic connected subgraph containing $n$ disjoint…

The Ramsey number $R(s,t)$ is the least integer $n$ such that any coloring of the edges of $K_n$ with two colors produces either a monochromatic $K_s$ in one color or a monochromatic $K_t$ in the other. If $s=t$, we say that the Ramsey…

For a positive integer $r$, let $G(r)$ be the smallest $N$ such that, whenever the edges of the Cartesian product $K_N \times K_N$ are $r$-coloured, then there is a rectangle in which both pairs of opposite edges receive the same colour. In…

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…

The odd-Ramsey number $r_{\text{odd}}(n,H)$ of a graph $H$ is the minimum number of colors needed to edge-color $K_n$ so that in every copy of $H$ some color occurs an odd number of times, and the unique-Ramsey number $r_{\text{u}}(n,H)$ is…

Given a graph $G$ and a real $\varepsilon>0$, an edge-coloring of $G$ is called $\varepsilon$-balanced if each color appears on at least an $\varepsilon$-fraction of the edges in $G$. A classical result of Erd\H{o}s and Szemer\'{e}di…

A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations $E$ there is a threshold value $R_k(E)$ (the Rado number of $E$) such that for any $k$-coloring of the integers in the interval $[1,n]$, with $n \ge…

Erd\H{o}s and Rado [P. Erd\H{o}s, R. Rado, A combinatorial theorem, Journal of the London Mathematical Society 25 (4) (1950) 249-255] introduced the Canonical Ramsey numbers $\text{er}(t)$ as the minimum number $n$ such that every…

Given a hypergraph $G$ and a subhypergraph $H$ of $G$, the \emph{odd Ramsey number} $r_{odd}(G,H)$ is the minimum number of colors needed to edge-color $G$ so that every copy of $H$ intersects some color class in an odd number of edges.…

The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ such that every red-blue coloring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of $G$. Determining or estimating these numbers is one of the…

We prove that for all graphs with at most $(3.75-o(1))n$ edges there exists a 2-coloring of the edges such that every monochromatic path has order less than $n$. This was previously known to be true for graphs with at most $2.5n-7.5$ edges.…

It was previously shown that any two-colour colouring of K(C_n) must contain a monochromatic planar K_4 subgraph for n >= N^*, where 6 <= N^* <= N and N is Graham's number. The bound was later improved to 11 <= N^* <= N. In this article, it…

We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a {\em set} of colours (instead of just one colour). We give bounds for monochromatic…

We consider following geometric Ramsey problem: find the least dimension $n$ such that for any 2-coloring of edges of complete graph on the points $\{\pm 1\}^n$ there exists 4-vertex coplanar monochromatic clique. Problem was first analyzed…

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…