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Towards a high-dimensional Dirac's theorem

Hyunwoo Lee

Abstract

Dirac's theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering hypergraph matchings and Hamiltonian cycles. In this paper, we consider another natural generalization of perfect matchings, Steiner triple systems. As a Steiner triple system can be viewed as a partition of pairs of vertices, it is a natural high-dimensional analogue of a perfect matching in graphs. We prove that for sufficiently large integer $n$ with $n \equiv 1 \text{ or } 3 \pmod{6}$, any $n$-vertex $3$-uniform hypergraph $H$ with minimum codegree at least $\left(\frac{3 + \sqrt{57}}{12} + o(1) \right)n = (0.879... + o(1))n$ contains a Steiner triple system. In fact, we prove a stronger statement by considering transversal Steiner triple systems in a collection of hypergraphs. We conjecture that the number $\frac{3 + \sqrt{57}}{12}$ can be replaced with $\frac{3}{4}$ which would provide an asymptotically tight high-dimensional generalization of Dirac's theorem.

Towards a high-dimensional Dirac's theorem

Abstract

Dirac's theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering hypergraph matchings and Hamiltonian cycles. In this paper, we consider another natural generalization of perfect matchings, Steiner triple systems. As a Steiner triple system can be viewed as a partition of pairs of vertices, it is a natural high-dimensional analogue of a perfect matching in graphs. We prove that for sufficiently large integer with , any -vertex -uniform hypergraph with minimum codegree at least contains a Steiner triple system. In fact, we prove a stronger statement by considering transversal Steiner triple systems in a collection of hypergraphs. We conjecture that the number can be replaced with which would provide an asymptotically tight high-dimensional generalization of Dirac's theorem.
Paper Structure (20 sections, 17 theorems, 32 equations, 1 figure)

This paper contains 20 sections, 17 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Combinatorial designs
  3. Transversals
  4. Fractional relaxations
  5. Our results
  6. Proof sketch.
  7. Organization.
  8. Preliminaries
  9. General notations
  10. Pseudorandom hypergraphs
  11. Probabilistic inequalities
  12. Fractional thresholds
  13. Pseudorandom structures and exclusive absorbers
  14. Pseudorandom fractional STS
  15. Exclusive absorbers
  16. ...and 5 more sections

Key Result

Theorem 1.2

For any positive real number $\varepsilon > 0$, there is $n_0 = n_0(\varepsilon)$ that satisfies the following for all $n \geq n_0$ satisfying $n\equiv 1\text{ or } 3 \pmod{6}$. If $H$ is an $n$-vertex $3$-uniform hypergraph where the minimum codegree of $H$ is at least $\mathopen{}\mathclose{\left(

Figures (1)

  • Figure 1: Vertex subsets and edge subsets of $H$.

Theorems & Definitions (45)

  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4: Barber et al. barber2016edge
  • Definition 1.5
  • Theorem 1.6
  • Proposition 1.7
  • proof
  • Theorem 1.8
  • Corollary 1.9
  • Definition 2.1
  • ...and 35 more