Two Results on Ramsey-Turán Theory

Abstract

Let $f(n)$ be a positive function and $H$ a graph. Denote by $\textbf{RT}(n,H,f(n))$ the maximum number of edges of an $H$-free graph on $n$ vertices with independence number less than $f(n)$. It is shown that  $\textbf{RT}(n,K_4+mK_1,o(\sqrt{n\log n}))=o(n^2)$ for any fixed integer $m\geqslant 1$ and $\textbf{RT}(n,C_{2m+1},f(n))=O(f^2(n))$ for any fixed integer $m\geqslant 2$ as $n\to\infty$.