Two Ramsey-Tur\'{a}n numbers involving triangles

TL;DR

This paper determines specific Ramsey-Turán densities for certain triangle and larger clique configurations, resolving long-standing open problems and characterizing the structure of extremal graphs.

Contribution

It computes exact values of  ho(3,6) and  ho(3,7), and shows these extremal graphs are weakly stable, addressing a 30-year-old open problem.

Findings

 ho(3,6) = 5/12

 ho(3,7) = 7/16

Extremal structures are weakly stable

Abstract

Given integers $p, q\ge2$, we say that a graph $G$ is $(K_p,K_q)$-free if there exists a red/blue edge coloring of $G$ such that it contains neither a red $K_p$ nor a blue $K_q$. Fix a function $f( n )$, the Ramsey-Tur\'{a}n number $RT( {n,p,q,f( n ))} $ is the maximum number of edges in an $n$-vertex $(K_p,K_q)$-free graph with independence number at most $f( n )$. For any $\delta>0$, let $\rho (p, q,\delta ) = \mathop {\lim }\limits_{n \to \infty } \frac{RT(n,p, q,\delta n)}{n^2}$. We always call $\rho (p, q):= \mathop {\lim }\limits_{\delta \to 0}\rho (p, q,\delta )$ the Ramsey-Tur\'{a}n density of $K_p$ and $K_q$. In 1993, Erd\H{o}s, Hajnal, Simonovits, S\'{o}s and Szemer\'{e}di proposed to determine the value of $\rho(3,q)$ for $q\ge3$, and they conjectured that for $q \ge 2$, $\rho \left( {3,2q - 1} \right) = \frac{1}{2}(1 - \frac{1}{r(3,q) - 1})$. Recently, Kim, Kim and Liu (2019) conjectured that for $q \ge 2$, $\rho( {3,2q } ) = \frac{1}{2}( 1 - \frac{1}{r( {3,q} )})$. Erd\H{o}s et al. (1993) determined $\rho(3,q)$ for $q=3,4,5$ and $\rho(4,4)$. There is no progress on the Ramsey-Tur\'{a}n density $\rho (p, q)$ in the past thirty years. In this paper, we obtain $\rho(3,6)=\frac{5}{12}$ and $\rho(3,7)=\frac{7}{16}$. Moreover, we show that the corresponding asymptotically extremal structures are weakly stable, which answers a problem of Erd\H{o}s et al. (1993) for the two cases.