Description
Given a graph $F$, a Berge copy of $F$ (Berge-$F$ for short) is a hypergraph obtained by enlarging the edges arbitrarily. Gy\H{o}ri, Salia and Zamora [\textit{European J. Combin.} 96 (2021) 103353] determined the maximum number of hyperedges in a connected $r$-uniform hypergraph on $n$ vertices containing no Berge path of length $k-1$ for $k\geq 2r+14$ and sufficiently large $n$, and asked for the minimum $k_0$ such that this extremal number holds for all $k\geq k_0$.
In this paper, we prove that the extremal number holds for all $k\geq 2r+2$ and fails for $k\le 2r+1$, thereby completely resolving the problem posed by Gyori, Salia and Zamora.
Moreover, we also improve the result of F\"uredi, Kostochka and Luo [\textit{Electron. J. Comb.} 26(4) (2019) 4--31], who determined the maximum number of hyperedges in a $2$-connected $n$-vertex $r$-uniform hypergraph containing no Berge cycle of length at least $k$ for $k\geq 4r$ and sufficiently large $n$, by showing that this extremal number holds for all $k\geq 2r+2$ and fails for $k\le 2r+1$.
Our approach reduces the Berge-Tur\'an problem to a graph extremal problem, and applies recent work of Ai, Lei, Ning and Shi [\textit{Canad. J. Math.} (2025) 1--27] on the feasibility of graph parameters and the Kelmans operation.
This is a joint work with Daniel Gerbner and Junpeng Zhou
Zoom:
Meeting ID: 895 2960 8626
Passcode: 627606
Link: https://us06web.zoom.us/j/89529608626?pwd=Y4YMgg9b3QvdPmbym7JPMTvyNMpPwb.1