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Triple systems with bounded matching number: some constructions and exact Turán number

Nannan Chen, Miao Liu, Yuzhen Qi, Caihong Yang

TL;DR

This work addresses Turán-type extremal problems for 3-graphs under joint constraints of forbidding a fixed F and a matching $M_{s+1}^3$. It develops an edge-colored Turán framework to derive asymptotic bounds, constructs infinite counterexamples to a conjecture when $\chi(F)=2$, and identifies exact extremal structures in key regimes, notably for $F_{3,2}$ when $n\ge 12s^2$. It also demonstrates a positive case via the $J_t$ family and discusses the limits of $q(F)=\infty$-based predictions, highlighting nuanced behavior across families. The results advance understanding of how colorings and neighborhood structure influence Turán numbers under matching restrictions and raise new questions about exact values and potential generalizations.

Abstract

We study the Turán numbers of $3$-graphs avoiding $3$-graphs $F$ and $M_{s+1}^3$, a matching of size $s+1$. We disprove a conjecture of Gerbner, Tompkins, and Zhou [European Journal of Combinatorics, 2025, 127:104155] on $\ex(n,\{F,M^3_{s+1}\})$ for $3$-graph $F$ with $χ(F)=2$ by constructing infinitely many counterexamples. For this family, we determine the asymptotic Turán number via edge-colored Turán problem. In addition, for the $3$-graph $F_{3,2}$ with edge set $\{123,145,245,345\}$, we determine the exact value of $\ex(n,\{F_{3,2}, M_{s+1}^3\})$ for every integers $s$ and all $n \ge 12s^2$.

Triple systems with bounded matching number: some constructions and exact Turán number

TL;DR

This work addresses Turán-type extremal problems for 3-graphs under joint constraints of forbidding a fixed F and a matching . It develops an edge-colored Turán framework to derive asymptotic bounds, constructs infinite counterexamples to a conjecture when , and identifies exact extremal structures in key regimes, notably for when . It also demonstrates a positive case via the family and discusses the limits of -based predictions, highlighting nuanced behavior across families. The results advance understanding of how colorings and neighborhood structure influence Turán numbers under matching restrictions and raise new questions about exact values and potential generalizations.

Abstract

We study the Turán numbers of -graphs avoiding -graphs and , a matching of size . We disprove a conjecture of Gerbner, Tompkins, and Zhou [European Journal of Combinatorics, 2025, 127:104155] on for -graph with by constructing infinitely many counterexamples. For this family, we determine the asymptotic Turán number via edge-colored Turán problem. In addition, for the -graph with edge set , we determine the exact value of for every integers and all .

Paper Structure

This paper contains 10 sections, 12 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Counterexamples and edge-colored Turán
  3. A positive case for Conjecture \ref{['conj1']}
  4. The Turán number of $\{F_{3,2},M_{s+1}^3\}$
  5. Preliminary
  6. Edge-colored Turán problem
  7. Counterexamples
  8. Proof of Theorem \ref{['thm:J_t']}
  9. Proof of \ref{['thm:f32']}
  10. Concluding Remark

Key Result

Theorem 1.2

Let $F$ be an $n$-vertex $r$-graph with $\chi(F)=2$ and $n \geq(2 s+1) r-s$. Then,

Figures (1)

  • Figure 1.1: Bipartite $3$-graphs $F_{\mathcal{P}, 3}$ and $F_{\mathcal{P}, 4}$. The red, blue and green edges belong to the link graph of red, blue and green vertices, respectively.

Theorems & Definitions (33)

  • Conjecture 1.1: Erdős Matching Conjecture, erdos1965
  • Theorem 1.2: gerbner2025
  • Conjecture 1.3: gerbner2025
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • ...and 23 more