Department of

March 2017 April 2017 May 2017 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 1 1 2 3 4 5 6 5 6 7 8 9 10 11 2 3 4 5 6 7 8 7 8 9 10 11 12 13 12 13 14 15 16 17 18 9 10 11 12 13 14 15 14 15 16 17 18 19 20 19 20 21 22 23 24 25 16 17 18 19 20 21 22 21 22 23 24 25 26 27 26 27 28 29 30 31 23 24 25 26 27 28 29 28 29 30 31 30

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.

Tuesday, January 31, 2017

Tuesday, February 7, 2017

Tuesday, February 14, 2017

Tuesday, February 21, 2017

Tuesday, February 28, 2017

Tuesday, March 7, 2017

Tuesday, March 14, 2017

Tuesday, April 11, 2017

Tuesday, April 18, 2017

Tuesday, April 25, 2017

Tuesday, August 29, 2017

Tuesday, September 5, 2017

Tuesday, September 19, 2017

Tuesday, September 26, 2017

Tuesday, October 3, 2017

Tuesday, October 10, 2017

Tuesday, October 17, 2017

Tuesday, October 24, 2017