The Turán Theorem for Random Graphs

The aim of this paper is to prove a Turan-type theorem for random graphs. For $\gamma >0$ and graphs $G$ and $H$, write $G\to_\gamma H$ if any $\gamma$-proportion of the edges of $G$ spans at least one copy of $H$ in $G$. We show that for every graph $H$ and every fixed real $\delta>0$, almost every graph $G$ in the binomial random graph model $\cG(n,q)$, with $q=q(n)\gg((\log n)^4/n)^{1/d(H)}$, satisfies $G\to_{(\chi(H)-2)/(\chi(H)-1)+\delta}H$, where as usual $\chi(H)$ denotes the chromatic number of $H$ and $d(H)$ is the ‘degeneracy number’ of $H$.Since $K_l$, the complete graph on $l$ vertices, is $l$-chromatic and $(l-1)$-degenerate, we infer that for every $l\geq2$ and every fixed real $\delta>0$, almost every graph $G$ in the binomial random graph model $\cG(n,q)$, with $q=q(n)\gg((\log n)^4/n)^{1/(l-1)}$, satisfies $G\to_{(l-2)/(l-1)+\delta}K_l$.