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 $\delta>0$, 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 $\rho(3,6,\delta)\ge \frac{ 5}{ 12}+\frac{ \delta}{ 2}+2\delta^2$ from a skilful construction and conjectured the equality holds for sufficiently small $\delta>0$. We make the first step to the conjecture by showing that $\rho(3,6,\delta)\le\frac{ 5}{ { 12}} + \frac{ \delta }{ 2}+ 2.1025\delta ^2$ for sufficiently small $\delta>0$.
Categories: math.CO