The Turán number of Berge paths in connected hypergraphs
Abstract
A Berge path in a hypergraph is an alternating sequence of distinct vertices and edges v1,e1, v2, …, ek−1, vk such that vi, vi+1 ∈ ei for all i ∈ [k−1]. Győri, Katona, and Lemons determined the maximum number of edges in r-uniform, n-vertex hypergraphs without Berge paths on k vertices. The extremal examples for their bounds are disconnected hypergraphs in which each component has either too few edges or too few vertices to contain such a Berge path. We consider the setting in which the host hypergraph is connected. Let exconnr(n,ℬ𝒫k) denote the maximum number of edges in an n-vertex, r-uniform, connected hypergraph that does not contain a Berge path on k vertices. This number was determined by Zhang, Broersma, Győri, Tompkins, Wang for k ≤ r+2, and Zhao, Gerbner, Zhou for k ≥ 2r+2. In this paper, we resolved the remaining cases r+3 ≤ k ≤ 2r+1 when n is sufficiently large, confirming a conjecture of Zhang, et al.
Related
- Turán number of a matching and a Berge triangle
- Saturation in three-uniform Berge hypergraphs: the five-star spectrum and the K4 saturation number
- Turán problems with bounded matching number in k-uniform hypergraphs
- Spectral extremal hypergraphs without long Berge cycles
- A Fan-Type Theorem for Hamiltonian Berge Cycles in 3-Uniform Hypergraphs
BibTeX
@misc{li2026turán,
title = {{The Turán number of Berge paths in connected hypergraphs}},
author = {Yupei Li and Ruth Luo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32875},
eprint = {2610.32875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32875},
}
RIS
TY - UNPB
TI - The Turán number of Berge paths in connected hypergraphs
AU - Yupei Li
AU - Ruth Luo
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.32875
UR - https://arxiv.org/abs/2610.32875
AB - A Berge path in a hypergraph is an alternating sequence of distinct vertices and edges v_1,e_1, v_2, …, e_k−1, v_k such that v_i, v_i+1 ∈ e_i for all i ∈ [k−1]. Győri, Katona, and Lemons determined the maximum number of edges in r-uniform, n-vertex hypergraphs without Berge paths on k vertices. The extremal examples for their bounds are disconnected hypergraphs in which each component has either too few edges or too few vertices to contain such a Berge path. We consider the setting in which the host hypergraph is connected. Let ex^conn_r(n,ℬ𝒫_k) denote the maximum number of edges in an n-vertex, r-uniform, connected hypergraph that does not contain a Berge path on k vertices. This number was determined by Zhang, Broersma, Győri, Tompkins, Wang for k ≤ r+2, and Zhao, Gerbner, Zhou for k ≥ 2r+2. In this paper, we resolved the remaining cases r+3 ≤ k ≤ 2r+1 when n is sufficiently large, confirming a conjecture of Zhang, et al.
ER -