On Turán-type problems for Berge matchings

TL;DR

This work resolves the remaining case $s\le r\le 2s+1$ for the Turán numbers of Berge matchings, giving exact and asymptotic formulas for $\mathrm{ex}_r(n, \mathcal{B}M_{s+1})$ and extending the analysis to Berge matchings together with a single $r$-graph and with Berge bipartite graphs. It introduces a unified approach for combining Berge matchings with additional forbidden structures, yielding explicit expressions such as $\mathrm{ex}_r(n, \mathcal{B}M_{s+1} \cup \{\mathcal{F}\}) = \mathrm{ex}_r(s, \mathcal{D}(\mathcal{F})) + {s \choose r-1}(n-s)$ under suitable chromaticity conditions, and derives asymptotics for bipartite and complete bipartite targets. The paper further generalizes Turán problems for families of Berge hypergraphs, providing upper bounds via red-blue colorings and extending prior results in the literature. Techniques rely on careful structural decomposition around maximum Berge matchings, two-type edge classifications, and Gallai–Edmonds-type tools to control forbidden configurations. Overall, the results advance the understanding of extremal behavior in Berge hypergraphs and illuminate the interaction between matchings and auxiliary hypergraphs beyond the bipartite setting.

Abstract

For a graph $F$, an $r$-uniform hypergraph ($r$-graph for short) $\mathcal{H}$ is a Berge-$F$ if there is a bijection $φ:E(F)\rightarrow E(\mathcal{H})$ such that $e\subseteq φ(e)$ for each $e\in E(F)$. Given a family $\mathcal{F}$ of $r$-graphs, an $r$-graph is $\mathcal{F}$-free if it does not contain any member in $\mathcal{F}$ as a subhypergraph. The Turán number of $\mathcal{F}$ is the maximum number of hyperedges in an $\mathcal{F}$-free $r$-graph on $n$ vertices. Kang, Ni, and Shan [\textit{Discrete Math. 345 (2022) 112901}] determined the exact value of the Turán number of Berge-$M_{s+1}$ for all $n$ when $r\leq s-1$ or $r\geq 2s+2$, where $M_{s+1}$ denotes a matching of size $s+1$. In this paper, we settle the remaining case $s\le r\le 2s+1$. Moreover, we establish several exact and general results on the Turán numbers of Berge matchings together with a single $r$-graph, as well as of Berge matchings together with Berge bipartite graphs. Finally, we generalize the results on Turán problems for Berge hypergraphs proposed by Gerbner, Methuku, and Palmer [\textit{Eur. J. Comb. 86 (2020) 103082}].

Figures (3)

• Figure 1: The Fano plane ${\mathcal{F}}_7$ and the hypergraph ${\mathcal{F}}_6$.
• Figure 2: The illustration of ${\mathcal{B}} M_{s+1}$ by the thick red lines. The left figure shows when a hyperedge $e \in {\mathcal{E}}_3$ contains $u_i, u_j$ (the figure shows the case $2 \le i,j \le s-1$, the case when $i \in \{1,s\}$ holds similarly). The right figure shows when a hyperedge $e\in {\mathcal{E}}_3$ contains $u_i, v_1$ ($2\leq i\leq s$). The hyperedge connecting $w'$ comes from the definition of $Y$.
• Figure 3: The four cases: $e_j$ contains $w$, $w" \neq w$, a vertex in $\overline{X}$, or a vertex in $\overline{Y}$. In each case, we can find a ${\mathcal{B}} M_{s+1}$ using either red lines or dotted lines. In the picture, we only list some typical cases. For example, when $e_j$ contains $u_2 \in \overline{X}$, we only consider $u_2 \in e_j\cap \overline{X}$. However, other cases hold similarly.

Theorems & Definitions (42)

• Theorem 1.1: Alon and Frankl ALON2024223
• Theorem 1.2: Gerbner Gerbner2024matching
• Theorem 1.3: Ma and Hou 2023arXiv230105625M
• Theorem 1.4: Khormali and Palmer KHORMALI2022103506
• Theorem 1.5: Kang, Ni and Shan KANG2022112901
• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Corollary 2.4
• Corollary 2.5