An explicit Ramsey graph

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.