Turán number of a matching and a Berge triangle
Abstract
For a fixed graph G, an r-uniform hypergraph is said to contain a Berge-G if there exists a bijection f: E(G)→ E(ℋ) for some subhypergraph ℋ such that e⊆ f(e) for every e∈ E(G). Motivated by Alon and Frankl's study of Turán problems under bounded matching constraints, we investigate the maximum number of edges in r-uniform Berge-K3-free hypergraphs with matching number at most~s. We determine the exact Turán numbers for the cases r=3 and r=4. For r=3 and n ≥ 3 s, we prove that every n-vertex Berge- K3-free 3-graph with matching number s has at most $s(n-2 s)$ edges, and we characterize the unique extremal hypergraph attaining equality. For r=4 and n ≥ 4 s, the maximum number of edges is s⌊(n−2 s) / 2⌋, except for the exceptional case s=1 and n ≡ 1(bmod 4), in which the bound is $(n-1) / 2$. As a corollary, our results recover the classical theorem of Győri on Berge-K3-free hypergraphs.
Related
- Hypergraph Turán Problem of the Generalized Triangle with Bounded Matching Number
- The Turán number of Berge paths in connected hypergraphs
- Turán problems with bounded matching number in k-uniform hypergraphs
- Random independent sets in triangle-free and linear Berge-C4-free hypergraphs
- Saturation in three-uniform Berge hypergraphs: the five-star spectrum and the K4 saturation number
BibTeX
@misc{xu2026turán,
title = {{Turán number of a matching and a Berge triangle}},
author = {Yue Xu and Jiasheng Zeng and Xiao-Dong Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.12300},
eprint = {2607.12300},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.12300},
}
RIS
TY - UNPB
TI - Turán number of a matching and a Berge triangle
AU - Yue Xu
AU - Jiasheng Zeng
AU - Xiao-Dong Zhang
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2607.12300
UR - https://arxiv.org/abs/2607.12300
AB - For a fixed graph G, an r-uniform hypergraph is said to contain a Berge-G if there exists a bijection f: E(G)→ E(ℋ) for some subhypergraph ℋ such that e⊆ f(e) for every e∈ E(G). Motivated by Alon and Frankl's study of Turán problems under bounded matching constraints, we investigate the maximum number of edges in r-uniform Berge-K_3-free hypergraphs with matching number at most~s. We determine the exact Turán numbers for the cases r=3 and r=4. For r=3 and n ≥ 3 s, we prove that every n-vertex Berge- K_3-free 3-graph with matching number s has at most $s(n-2 s)$ edges, and we characterize the unique extremal hypergraph attaining equality. For r=4 and n ≥ 4 s, the maximum number of edges is s⌊(n−2 s) / 2⌋, except for the exceptional case s=1 and n ≡ 1(bmod 4), in which the bound is $(n-1) / 2$. As a corollary, our results recover the classical theorem of Győri on Berge-K_3-free hypergraphs.
ER -