Extremal results on Berge disjoint paths

TL;DR

This paper determines the exact Turán number for Berge disjoint paths in hypergraphs, extending previous results to all cases with multiple disjoint paths of certain lengths.

Contribution

It provides a comprehensive formula for the Turán number of Berge disjoint paths for all parameters with hypergraph uniformity at least 3 and path length at least r+7.

Findings

Exact Turán number for Berge disjoint paths derived

Generalizes previous results to all disjoint path cases

Extends understanding of Berge path extremal problems

Abstract

The well-known Erd\H{o}s-Gallai Theorem gave the Tur\'an number of paths. Bushaw and Kettle generalized this result to consider the Tur\'an number of disjoint paths. Since then, many studies are focused on the Tur\'an number of linear forest. For a graph $F$, an $r$-uniform hypergraph $\mathcal{H}$ is a $\text{Berge-} F$ if there is a bijection $\phi: E(F)\to E(\mathcal{H})$ such that $e\subseteq \phi(e)$ for each $e\in E(F)$. When $F$ is a path, we call $\text{Berge-} F$ a Berge path. The Tur\'an number of Berge paths was initially studied by Gy\H{o}ri, Katona and Lemons. They gave the value of $\text{ex}_r(n,\text{Berge-}P_\ell)$ for $\ell>r+1$. This result is a generalization of Erd\H{o}s-Galli Theorem. Since then, the Tur\'an number of Berge paths has received widespread attention. Recently, Zhou, Gerbner and Yuan initially studied the Tur\'an number of Berge disjoint paths and for the cases when all the paths have odd length. In this paper, we give a more general result, which gives the exact value of $\mathrm{ex}_r(n,\text{Berge-} kP_{\ell})$ for all $k\geq 2$, $r\ge 3$, and $\ell\geq r+7$.