The Ramsey-Tur\'{a}n problem for cliques

TL;DR

This paper determines the maximum edge density in large graphs avoiding both a complete subgraph of size t and a large independent set, providing explicit formulas for small delta depending on whether t is even or odd.

Contribution

It extends the understanding of the Ramsey-Turán problem by deriving exact asymptotic formulas for the maximum edge density for small delta, building on recent work.

Findings

Explicit formulas for f_t(δ) when δ is small, depending on parity of t.

Confirmation of the asymptotic behavior of extremal graphs in the Ramsey-Turán setting.

Advancement in extremal graph theory for large graphs avoiding cliques and large independent sets.

Abstract

An important question in extremal graph theory raised by Vera T. S\'os asks to determine for a given integer $t\ge 3$ and a given positive real number $\delta$ the asymptotically supremal edge density $f_t(\delta)$ that an $n$-vertex graph can have provided it contains neither a complete graph $K_t$ nor an independent set of size $\delta n$. Building upon recent work of Fox, Loh, and Zhao [The critical window for the classical Ramsey-Tur\'an problem, Combinatorica 35 (2015), 435-476], we prove that if $\delta$ is sufficiently small (in a sense depending on $t$), then \[ f_t(\delta)= \begin{cases} \frac{3t-10}{3t-4}+\delta-\delta^2 & \text{ if $t$ is even,} \cr \frac{t-3}{t-1}+\delta & \text{ if $t$ is odd.} \end{cases} \]