Geodesic
The maximum number of $P_\ell$ copies in $P_k$-free graphs
Discrete mathematics & theoretical computer science, ICGT 2018, Tome 21 (2019) no. 1.

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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. 
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     title = {The maximum number of $P_\ell$ copies in $P_k$-free graphs},
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Győri, Ervin; Salia, Nika; Tompkins, Casey; Zamora, Oscar. The maximum number of $P_\ell$ copies in $P_k$-free graphs. Discrete mathematics & theoretical computer science, ICGT 2018, Tome 21 (2019) no. 1. doi : 10.23638/DMTCS-21-1-14. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-1-14/

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