Critical behaviors of the Ramsey-Turán number of $K_3$ and $K_6$

Abstract: In 1969, Erd\H{o}s and S\'{o}s initiated the study of the Ramsey-Tur\'{a}n 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 $\delta&gt;0$, the Ramsey-Tur\'{a}n number $RT( {n,p,q,\delta n)}$ is the maximum number of edges in an $n$-vertex $(K_p,K_q)$-free graph with independence number at most $\delta n$. Let $$\rho (p, q,\delta ) = \mathop {\lim }\limits_{n \to \infty } \frac{RT(n,p, q,\delta n)}{n<sup>2}$$. Kim, Kim and Liu (2019) showed $$\rho(3,6,\delta)\ge \frac{5}{12}+\frac{\delta}{2}+2\delta<sup>2$$ from a skilful construction and conjectured the equality holds for sufficiently small $\delta&gt;0$. We make the first step to the conjecture by showing that $$\rho(3,6,\delta)\le\frac{5}{{12}} + \frac{\delta }{2}+ 2.1025\delta <sup>2$$ for sufficiently small $\delta&gt;0$.