On the Ramsey-Tur\'an numbers of graphs and hypergraphs

TL;DR

This paper investigates the maximum edges in large graphs avoiding certain cliques and induced subgraphs, providing new bounds and constructions that advance understanding of Ramsey-Turán numbers for graphs and hypergraphs.

Contribution

It proves that the Ramsey-Turán number RT_t(n, K_{t+2}, o(n)) grows quadratically, answering a key open question and extending results to hypergraphs.

Findings

RT_t(n, K_{t+2}, o(n)) = Ω(n^2)

Constructs imply new hypergraph Ramsey-Turán results

Answers a long-standing open problem in extremal graph theory

Abstract

Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Tur\'an number of H, RT_t(n, H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G where f(n) is larger than the maximum number of vertices in a $K_t$-free induced subgraph of G. Erd\H{o}s, Hajnal, Simonovits, S\'os, and Szemer\'edi posed several open questions about RT_t(n,K_s,o(n)), among them finding the minimum s such that $RT_t(n,K_{t+s},o(n)) = \Omega(n^2)$, where it is easy to see that $RT_t(n,K_{t+1},o(n)) = o(n^2)$. In this paper, we answer this question by proving that $RT_t(n,K_{t+2},o(n)) = \Omega(n^2)$; our constructions also imply several results on the Ramsey-Tur\'an numbers of hypergraphs.