Ramsey graphs induce subgraphs of many different sizes

TL;DR

This paper proves that $C$-Ramsey graphs on $n$ vertices contain a vast number of subgraphs with distinct sizes, supporting the idea that such graphs share properties with dense random graphs.

Contribution

It establishes that $C$-Ramsey graphs induce at least $n^{2-o(1)}$ subgraph sizes, linking to longstanding conjectures in graph theory.

Findings

$C$-Ramsey graphs have many subgraph sizes

The result is near-optimal and related to open conjectures

Supports the similarity between Ramsey and dense random graphs

Abstract

A graph on $n$ vertices is said to be \emph{$C$-Ramsey} if every clique or independent set of the graph has size at most $C \log n$. The only known constructions of Ramsey graphs are probabilistic in nature, and it is generally believed that such graphs possess many of the same properties as dense random graphs. Here, we demonstrate one such property: for any fixed $C>0$, every $C$-Ramsey graph on $n$ vertices induces subgraphs of at least $n^{2-o(1)}$ distinct sizes. This near-optimal result is closely related to two unresolved conjectures, the first due to Erd\H{o}s and McKay and the second due to Erd\H{o}s, Faudree and S\'{o}s, both from 1992.