The Turán 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 $1\le i\le k$. Győri, Katona and Lemons [\textit{European J. Combin. 58 (2016) 238--246}] generalized the Erdős-Gallai theorem to Berge paths and established bounds for the Turán 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 Turán 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 Turán number of Berge paths.

21 pages