Generalized Ramsey-Tur\'an Numbers

TL;DR

This paper extends the understanding of the maximum number of larger cliques in graphs that avoid certain complete subgraphs and have small independence number, generalizing previous results for triangles to larger cliques.

Contribution

It establishes asymptotic results for counting copies of larger cliques in $K_p$-free graphs with small independence number, and provides counterexamples to a related conjecture.

Findings

Asymptotics for $K_4$ and $K_5$ copies in $K_p$-free graphs with small independence number.

Results for the case $p extgreater= 5q$, extending previous work.

Counterexamples to a conjecture by Balogh, Liu, and Sharifzadeh.

Abstract

The Ramsey-Tur\'an problem for $K_p$ asks for the maximum number of edges in an $n$-vertex $K_p$-free graph with independence number $o(n)$. In a natural generalization of the problem, cliques larger than the edge $K_2$ are counted. Let {\bf RT}$(n,\#K_q,K_p,o(n))$ denote the maximum number of copies of $K_q$ in an $n$-vertex $K_p$-free graph with independence number $o(n)$. Balogh, Liu and Sharifzadeh determined the asymptotics of {\bf RT}$(n,\# K_3,K_p,o(n))$. In this paper we will establish the asymptotics for counting copies of $K_4$, $K_5$, and for the case $p \geq 5q$. We also provide a family of counterexamples to a conjecture of Balogh, Liu and Sharifzadeh.