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.
Categories: math.CO