Let KL denote König's Lemma (for trees over $\mathbb{N}$), and RT(3) denote the Infinite Ramsey Theorem for triples over $\mathbb{N}$ (notation as in Simpson's book Subsystems …

I am interested in graph Ramsey theory. Are there any papers which investigate Ramsey numbers $R(G,G)$ of an arbitrary graph by analyzing the spectrum of $G$? In general, has anyon …

Consider positive integers $c$, $k$, and $s$. Does there exist some $N = N(c,k,s)$ such that the following holds? Take any $c$-coloring of the $k$-tuples of integers in $[1,N] …

Choose $k$. Let $G = (V,E)$ be a graph on $n = R(k,k)-1$ vertices (that is, $G$ is an extremal example for $R(k,k)$, and $g : E \to {r, b}$ be an edge 2-coloring such that there is …

Can anyone explain me what is a Ramsey Graph with a simple example? What are its properties?

Define a length $k$ arithmetic progression in $\mathbb{Z}_p$ to be a set of the form $\{ax+b : x \in [k]\}$ with $a \in \mathbb{Z}_p^*$ and $b \in \mathbb{Z}_p$. Let $HJ(k, c)$ be …

Graham's number achieved a kind of cult status, thanks to Martin Gardner, as the largest finite number appearing in a mathematical proof. (It may no longer hold that record, but th …

There are few results in modern mathematics that I find so deep and full of philosophical implications as Ramsey's theorem. I am aware (at some basic level) that it has generated …

There is a well-known fact in infinite combinatorics asserting that for each infinite linear order $P$ there is a countable subset $R\subseteq P$ of order type either $\omega$ or $ …

I'm studying about Graph Ramsey Theory now. Starting this study, I'm reading Chvatal and Harary's series of papers. In the second paper (V.Chvatal, F.Harary, Generalized ramsey the …

I am currently reading the recent preprint of Dodos, Kanellopoulos, Tyros, where the ambitiously short proof of Density Hales Jewett theorem is provided. The important ingredient i …

I assume the following Lemma is either well known or, more probably, a Corollary of a much stronger well known Theorem, and I would be grateful for a reference: For all $\delta\in …

Does there exist a graph $G$ which cannot be properly vertex-coloured with 3 colours (i.e. $G$ has chromatic number at least 4), such that for every graph $H$, if $H$ contains a …

I wonder what interesting and non-trivial examples of density Ramsey theorems with explicit asymptotics are there? I'm aware of two examples: Szemerédi's theorem and density Hales …

Finite Ramsey's theorem is a very important combinatorial tool that is often used in mathematics. The infinite version of Ramsey's theorem (Ramsey's theorem for colorings of tuples …