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- W2786975608 abstract "For graphs $T, H$, let $ex(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex $H$-free graph. In this paper we prove some sharp results on this generalization of Turan numbers, where our focus is for the graphs $T,H$ satisfying $chi(T) m$, Alon and Shikhelman showed that $ex(n,K_m,H)=binom{r}{m}(frac{n}{r})^m+o(n^m)$. Here we determine this error term $o(n^m)$ up to a constant factor. We prove that $ex(n,K_m,H)=binom{r}{m}(frac{n}{r})^m+biex(n,H)cdotTheta(n^{m-2})$, where $biex(n,H)$ is the Turan number of the decomposition family of $H$. As a special case, we extend Erdős' result, by showing that $T_r(n)$ uniquely attains $ex(n,K_m,H)$ for any edge-critical graph $H$. We also consider $T$ being non-clique, where even the simplest case seems to be intricate. Following from a more general result, we show that for all $sleq t$, $T_2(n)$ maximizes the number of $K_{s,t}$ in $n$-vertex triangle-free graphs if and only if $t<s+frac12+sqrt{2s+frac14}$." @default.
- W2786975608 created "2018-02-23" @default.
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- W2786975608 date "2018-02-04" @default.
- W2786975608 modified "2023-09-25" @default.
- W2786975608 title "Some sharp results on the generalized Tur'an numbers" @default.
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