Critical behaviors of the Ramsey-Turán number of $K_3$ and $K_6$
Xinyu Hu, Qizhong Lin
TL;DR
This paper advances the Ramsey-Turán program for two-color graphs by establishing a sharp upper bound on the Ramsey-Turán density $\rho(3,6,\delta)$ for sufficiently small $\delta>0$, showing $\rho(3,6,\delta) \le \frac{5}{12} + \frac{\delta}{2} + 2.1025\delta^{2}$. The approach combines a weak stability result with a new colored stability framework, producing a six-part partition and a reduced-weighted graph that must resemble the Turán graph $T_{m,6}$. By analyzing color patterns and leveraging regularity-based counting arguments, the authors rule out configurations that would violate the $(K_3,K_6)$-free condition, thereby tightening the upper bound toward the conjectured equality $\rho(3,6,\delta)=\frac{5}{12}+\frac{\delta}{2}+2\delta^{2}$ for small $\delta$. The work also outlines implications for $\rho(3,7,\delta)$ and presents a lower-bound construction suggesting a conjectured exact value $\rho(3,7,\delta)=\frac{7}{16}+\frac{\delta}{2}$, signaling directions for future research in multicolor Ramsey–Turán densities.
Abstract
In 1969, Erdős and Sós initiated the study of the Ramsey-Turán type problems. Given integers $p, q\ge2$, 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 $K_q$. For any $δ>0$, the Ramsey-Turán number $RT( {n,p,q,δn)} $ is the maximum number of edges in an $n$-vertex $(K_p,K_q)$-free graph with independence number at most $δn$. Let $ρ(p, q,δ) = \mathop {\lim }\limits_{n \to \infty } \frac{RT(n,p, q,δn)}{n^2}$. Kim, Kim and Liu (2019) showed $ρ(3,6,δ)\ge \frac{5}{12}+\fracδ{2}+2δ^2$ from a skilful construction and conjectured the equality holds for sufficiently small $δ>0$. We make the first step to the conjecture by showing that $ρ(3,6,δ)\le\frac{5}{12} + \frac{δ}{2}+ 2.1025δ^2$ for sufficiently small $δ>0$.