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

TL;DR

This paper determines the exact Turán number for Berge paths in r-uniform hypergraphs for the case where the path length exceeds the uniformity, completing previous partial results and resolving an open problem.

Contribution

It provides the final solution to the Turán number of Berge paths for all path lengths, especially when the path length exceeds the hypergraph's uniformity.

Findings

Exact Turán number for Berge paths when k > r is established.

Completes the classification of Turán numbers for Berge paths in hypergraphs.

Resolves an open problem in extremal hypergraph theory.

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.