A step towards the Ramsey-Tur\'{a}n conjecture for $K_3$ and $K_6$

Ramsey-Tur\'{a}n type problems were initiated by Erd\H{o}s and S\'{o}s in 1969. 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 $\delta>0$, 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 $\rho (p, q,\delta ) = \mathop {\lim }\limits_{n \to \infty } \frac{RT(n,p, q,\delta n)}{n^2}$. Kim, Kim and Liu (2019) showed that $\rho(3,6,\delta)\ge \frac{5}{12}+\frac{\delta}{2}+2\delta^2$ via a skillful construction and conjectured the equality holds for sufficiently small $\delta>0$. Using Szemer\'{e}di's regularity lemma and a stability argument, we make the first step towards the conjecture by showing that $\rho(3,6,\delta)$ is at most $\frac{5}{{12}} + \frac{\delta }{2}+ 2.1025\delta ^2$.