Proof of a conjecture on induced subgraphs of Ramsey graphs

TL;DR

This paper proves a long-standing conjecture that all C-Ramsey graphs contain at least on the order of n^{5/2} induced subgraphs with unique vertex-edge counts, highlighting their inherent combinatorial richness.

Contribution

The paper confirms the Erdős-Faudree-Sós conjecture, establishing a lower bound on the number of distinct induced subgraphs in C-Ramsey graphs, advancing understanding of their structure.

Findings

Proved that C-Ramsey graphs have at least Ω(n^{5/2}) distinct induced subgraphs.

Confirmed that no pair of these subgraphs share the same number of vertices and edges.

Extended previous results, showing inherent combinatorial richness of Ramsey graphs.

Abstract

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain "richness" properties characteristic of random graphs. More than 25 years ago, Erd\H{o}s, Faudree and S\'{o}s conjectured that in any C-Ramsey graph there are $\Omega\left(n^{5/2}\right)$ induced subgraphs, no pair of which have the same numbers of vertices and edges. Improving on earlier results of Alon, Balogh, Kostochka and Samotij, in this paper we prove this conjecture.