On the Ramsey-Tur\'an number with small $s$-independence number

TL;DR

This paper investigates the maximum edges in graphs avoiding certain cliques with small s-independence number, providing near-optimal bounds for fixed parameters and exploring phase transitions in this extremal problem.

Contribution

It establishes nearly tight bounds for the Ramsey-Turán number with small independence number for fixed parameters and extends understanding of phase transitions in these extremal graphs.

Findings

Proves lower bounds for $RT_s(n,K_{s+1}, n^{ ext{delta}})$ for large $s$ and $ ext{delta}$ between 1/2 and 1.

Shows these bounds are nearly optimal by matching trivial upper bounds asymptotically.

Discusses phase transitions in the behavior of $RT_s(n, K_{2s+1}, f)$ extending recent results.

Abstract

Let $s$ be an integer, $f=f(n)$ a function, and $H$ a graph. Define the Ramsey-Tur\'an number $RT_s(n,H, f)$ as the maximum number of edges in an $H$-free graph $G$ of order $n$ with $\alpha_s(G) < f$, where $\alpha_s(G)$ is the maximum number of vertices in a $K_s$-free induced subgraph of $G$. The Ramsey-Tur\'an number attracted a considerable amount of attention and has been mainly studied for $f$ not too much smaller than $n$. In this paper we consider $RT_s(n,K_t, n^{\delta})$ for fixed $\delta<1$. We show that for an arbitrarily small $\varepsilon>0$ and $1/2<\delta< 1$, $RT_s(n,K_{s+1}, n^{\delta}) = \Omega(n^{1+\delta-\varepsilon})$ for all sufficiently large $s$. This is nearly optimal, since a trivial upper bound yields $RT_s(n,K_{s+1}, n^{\delta}) = O(n^{1+\delta})$. Furthermore, the range of $\delta$ is as large as possible. We also consider more general cases and find bounds on $RT_s(n,K_{s+r},n^{\delta})$ for fixed $r\ge2$. Finally, we discuss a phase transition of $RT_s(n, K_{2s+1}, f)$ extending some recent result of Balogh, Hu and Simonovits.