The Tur\'{a}n number of Berge paths

Abstract

A Berge path of length $k$ in an $r$-uniform hypergraph is a collection of $k$ hyperedges $h_1,\dots,h_k$ and $k+1$ vertices $v_1,\dots,v_{k+1}$ such that $v_i, v_{i+1}\in h_i$ for each $1\le i\le k$. Gy\H{o}ri, Katona and Lemons [\textit{European J. Combin. 58 (2016) 238--246}] generalized the Erd\H{o}s-Gallai theorem to Berge paths and established bounds for the Tur\'{a}n number of Berge paths. However, these bounds are sharp only when some divisibility conditions hold. Gy\H ori, Lemons, Salia and Zamora [\textit{J. Combin. Theory Ser. B 148 (2021) 239--250}] determined the exact value of the Tur\'{a}n number of Berge paths in the case $k\le r$. In this paper, we settle the final open case $k>r$, thereby completing the determination of the Tur\'{a}n number of Berge paths.