Independence number of graphs with a prescribed number of cliques

TL;DR

This paper investigates the minimum independence number of graphs with a fixed number of s-cliques, revealing a phase transition at a specific clique count and providing sharp bounds for triangle-rich graphs.

Contribution

It establishes a phase transition in the independence number based on clique count and generalizes classical results in Ramsey theory for graphs with many triangles.

Findings

Identifies a transition point at t = n^{s/2+o(1)} for the independence number.

Provides sharp bounds for graphs with many triangles, extending Ramsey theory results.

Demonstrates the minimum independence number scales with the number of triangles and graph size.

Abstract

We consider the following problem posed by Erdos in 1962. Suppose that $G$ is an $n$-vertex graph where the number of $s$-cliques in $G$ is $t$. How small can the independence number of $G$ be? Our main result suggests that for fixed $s$, the smallest possible independence number undergoes a transition at $t=n^{s/2+o(1)}$. In the case of triangles ($s=3$) we obtain the following result which is sharp apart from constant factors and generalizes basic results in Ramsey theory: there exists $c>0$ such that every $n$-vertex graph with $t$ triangles has independence number at least $$c \cdot \min\left\{ \sqrt {n \log n}\, , \, \frac{n}{t^{1/3}} \left(\log \frac{n}{ t^{1/3}}\right)^{2/3} \right\}.$$