Related papers: Nearly tight exponents for off-diagonal Ramsey num…

In a recent breakthrough Campos, Griffiths, Morris and Sahasrabudhe obtained the first exponential improvement of the upper bound on the diagonal Ramsey numbers since 1935. We shorten their proof, replacing the underlying book algorithm…

We prove a new lower bound for the off-diagonal Ramsey numbers, \[ R(3,k) \geq \bigg( \frac{1}{3}+ o(1) \bigg) \frac{k^2}{\log k }\, , \] thereby narrowing the gap between the upper and lower bounds to a factor of $3+o(1)$. This improves…

For fixed $s \ge 3$, we prove that if optimal $K_s$-free pseudorandom graphs exist, then the Ramsey number $r(s,t) = t^{s-1+o(1)}$ as $t \rightarrow \infty$. Our method also improves the best lower bounds for $r(C_{\ell},t)$ obtained by…

We present a flexible random construction which, for certain graphs $H$, is able to produce $H$-free graphs with edge density strictly larger than that of the $H$-free process, while simultaneously preserving pseudorandom properties and…

Recently, Ma, Shen and Xie broke the Erd\H{o}s barrier for off-diagonal Ramsey numbers $R(\ell,C\ell)$, achieving the first exponential improvement over the classical lower bound for every $C>1$ and sufficiently large $\ell$. Hunter,…

We improve the upper bound for diagonal Ramsey numbers to \[R(k+1,k+1)\le\exp(-c(\log k)^2)\binom{2k}{k}\] for $k\ge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended…

The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that \[ R(k) \leqslant (4 -…

We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform Ramsey number $r_4(5,n)$, and the same for the iterated $(k-4)$-fold logarithm of the $k$-uniform…

The purpose of this survey is to provide a gentle introduction to several recent breakthroughs in graph Ramsey theory. In particular, we will outline the proofs (due to various groups of authors) of exponential improvements to the diagonal,…

The Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple…

The Ramsey number $r_k(s,n)$ is the minimum $N$ such that every red-blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ contains a red set of size $s$ or a blue set of size $n$, where a set is red (blue) if all of its $k$-subsets are red…

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. A well-known conjecture of Erd\H{o}s and Hajnal states that for any fixed…

The areas of Ramsey theory and random graphs have been closely linked ever since Erd\H{o}s' famous proof in 1947 that the 'diagonal' Ramsey numbers $R(k)$ grow exponentially in $k$. In the early 1990s, the triangle-free process was…

A graph is $(t_1, t_2)$-Ramsey if any red-blue coloring of its edges contains either a red copy of $K_{t_1}$ or a blue copy of $K_{t_2}$. The size Ramsey number is the minimum number of edges contained in a $(t_1,t_2)$-Ramsey graph.…

The set-colouring Ramsey number $R_{r,s}(k)$ is defined to be the minimum $n$ such that if each edge of the complete graph $K_n$ is assigned a set of $s$ colours from $\{1,\ldots,r\}$, then one of the colours contains a monochromatic clique…

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. We prove that $r_4(5,n)\ge 2^{2^{cn^{1/7}}}$, where $c>0$ is an absolute…

In this short note, we provide a new infinite family of $K_{2, t+1}$-free graphs for each prime power $t$. Using these graphs, we show that it is possible to partition the edges of $K_n$ into parts, such that each part is isomorphic to our…

We obtain some new upper bounds on the Ramsey numbers of the form $R(\underbrace{C_4,\ldots,C_4}_m,G_1,\ldots,G_n)$, where $m\ge 1$ and $G_1,\ldots,G_n$ are arbitrary graphs. We focus on the cases of $G_i$'s being complete, star $K_{1,k}$…

The $r$-colour Ramsey number $R_r(k)$ is the minimum $n \in \mathbb{N}$ such that every $r$-colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove, for each fixed $r \geqslant 2$,…

For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where…