Casey Tompkins: The maximum number of $P_\ell$ copies in $P_k$-free graphs

Generalizing Tur\'an's classical extremal problem, Alon and Shikhelman

investigated the problem of maximizing the number of $T$ copies in an

$H$-free graph, for a pair of graphs $T$ and $H$. Whereas Alon and

Shikhelman were primarily interested in determining the order of magnitude

for large classes of graphs $H$, we focus on the case when $T$ and $H$ are

paths, where we find asymptotic and in some cases exact results. We also

consider other structures like stars and the set of cycles of length at least $k$,

where we derive asymptotically sharp estimates. Our results generalize

well-known extremal theorems of Erd\H{o}s and Gallai.

This is joint work with Ervin Győri, Nika Salia and Oscar Zamora.