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Rényi, Nagyterem + Zoom
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Description

Abstract:

The $r$-uniform expansion $F^{(r)+}$ of a graph $F$ is obtained by enlarging each edge with $r-2$
new vertices such that altogether we use $(r-2)|E(F)|$ new vertices. Two simple lower bounds on the
largest number $\ex_r(n,F^{(r)+})$ of $r$-edges in $F^{(r)+}$-free $r$-graphs are $\Omega(n^{r-1})$
(in the case $F$ is not a star) and $\ex(n,K_r,F)$, which is the largest number of $r$-cliques in
$n$-vertex $F$-free graphs. We prove that $\ex_r(n,F^{(r)+})=\ex(n,K_r,F)+O(n^{r-1})$.
The proof comes with a structure theorem that we use to determine $\ex_r(n,F^{(r)+})$ exactly
for some graphs $F$, every $r<\chi(F)$ and sufficiently large $n$.

 

 

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