A conjecture of Erd\H{o}s on graph Ramsey numbers

TL;DR

This paper proves Erdős's longstanding conjecture that the Ramsey number of any graph with m edges and no isolated vertices is at most exponential in the square root of m, confirming a key hypothesis in combinatorics.

Contribution

The paper establishes Erdős's conjecture, showing that the Ramsey number for such graphs is bounded by an exponential function of the square root of the number of edges.

Findings

Proves Erdős's conjecture on graph Ramsey numbers.

Shows that r(G) ≤ 2^{c√m} for some absolute constant c.

Confirms a fundamental hypothesis in Ramsey theory.

Abstract

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 central problems in combinatorics. One of the oldest results in Ramsey Theory, proved by Erd\H{o}s and Szekeres in 1935, asserts that the Ramsey number of the complete graph with $m$ edges is at most $2^{O(\sqrt{m})}$. Motivated by this estimate Erd\H{o}s conjectured, more than a quarter century ago, that there is an absolute constant $c$ such that $r(G) \leq 2^{c\sqrt{m}}$ for any graph $G$ with $m$ edges and no isolated vertices. In this short note we prove this conjecture.