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Tuesday, January 24, 2017

**Abstract:** For two fixed graphs $T$ and $H$ let $ex(G(n,p),T,H)$ be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n,p)$. We show that for the case $T=K_m$ and $\chi(H)>m$ the behavior of $ex(G(n,p),K_m,H)$ depends strongly on the relation between $p$ and $m_2(H)=\max_{H'\subset H, |V(H')|\geq 3}\left\{ \frac{e(H')-1}{v(H')-2} \right\}$. When $m_2(H)>m_2(K_m)$ we prove that with high probability, depending on the value of $p$, either one can keep almost all copies of $K_m$ in an $H$-free subgraph of $G(n,p)$, or it is asymptotically best to take a $\chi(H)-1$ partite subgraph of $G(n,p)$. The transition between these two behaviors occurs at $p=n^{-1/m_2(H)}$. When $m_2(H)< m_2(K_m)$, the above cases still exist, however for $\delta>0$ small at $p=n^{-1/m_2(H)+\delta}$ one can typically still keep most of the copies of $K_m$. The reason for this is that although $K_m$ has the minimum average degree among the $m$-color-critical graphs, it does not have the smallest $m_2(G)$ among such graphs. This is joint work with N. Alon and C. Shikhelman.

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