An explicit Ramsey graph

TL;DR

This paper presents an explicit construction of Ramsey graphs that significantly improves the size of graphs avoiding large cliques and independent sets, using polynomial subspace methods and character theory.

Contribution

It introduces a new explicit construction of Ramsey graphs with larger size bounds than previous methods, advancing the understanding of explicit Ramsey graph construction.

Findings

Constructs graphs with at least m^{(1+o(1))(1/3)(log m / log log m)} vertices

Graphs contain no clique or independent set of size m

Uses polynomial subspace method and character theory of symmetric groups

Abstract

Explicit construction of Ramsey graphs has remained a challenging open problem for a long time. Frankl--Wilson \cite{FW}, Alon \cite{A} and Grolmusz \cite{G2} gave the best explicit constructions of graphs on $m$ vertices with no clique or independent set of size $m^{(1+o(1))\frac{1}{4}\frac{\log m}{\log \log m}}$. We describe here an explicit construction which produces for every integer $m>1$ a graph on at least $m^{(1+o(1))\frac{1}{3}\frac{\log m}{\log \log m}}$ vertices containing neither a clique of size $m$ nor an independent set of size $m$. In the proof we use the polynomial subspace method and some character theory of the complete symmmetric group.