Graphs with few 3-cliques and 3-anticliques are 3-universal

TL;DR

This paper proves that large graphs with few 3-cliques and 3-anticliques necessarily contain all 3-vertex subgraphs, establishing a sharp bound, and extends similar results to tournaments with 4-vertex subgraphs.

Contribution

It establishes the first sharp bounds for the presence of all 3-vertex subgraphs in graphs with few 3-cliques and 3-anticliques, and extends the phenomenon to tournaments.

Findings

Large graphs with few 3-cliques and 3-anticliques contain all 3-vertex subgraphs.

A sharp bound is proven for this property.

Similar results are shown for tournaments with 4-vertex subgraphs.

Abstract

For given integers k, l we ask whether every large graph with a sufficiently small number of k-cliques and k-anticliques must contain an induced copy of every l-vertex graph. Here we prove this claim for k=l=3 with a sharp bound. A similar phenomenon is established as well for tournaments with k=l=4.