Xin-yu HU, Qi-zhong LIN
Given a forbidden graph $H$ and a function $f(n)$, the Ramsey-Turán number $\textbf{RT}\left( {n,H,f\left( n \right)} \right)$ is the maximum number of edges of an $H$-free graph on $n$ vertices with independence number less than ${f\left( n \right)}$. For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any red/blue edge coloring of the complete graph $K_N$ contains either a red $G$ or a blue $H$. Denote $G+H$ by the join graph obtained from disjoint $G$ and $H$ by adding all edges between them completely. We first show that for any fixed graph $H$, if there are two constants $p:=p(H)>0$ and $q:=q(H)>1$ such that $R(H,K_n)\le \frac{pn^q}{(\log n)^{q-1}}$, then $\textbf{RT}(n,K_2+H,o(n^{\frac{1}{q}}(\log n)^{1-\frac{1}{q}}))=o(n^2),$ which extends several previous results. Moreover, we show that for any fixed forest $F$ of order $k\ge3$, and for any $0<\delta<1$ and sufficiently large $n$, \begin{align*} \textbf{RT}( {n,F+F,n^\delta} )\le n^{2-(1-\delta)/\lceil\frac{(k-1)(2-\delta)}{1-\delta}\rceil}. \end{align*} As a corollary, we have an upper bound for ${\bf{RT}}( {n,K_{2,2,2},n^{\delta}})$ for any $0<\delta<1$.
