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Codegree conditions for (fractional) Steiner triple systems

Michael Zheng

TL;DR

This work advances the threshold theory for Steiner triple systems in 3-uniform hypergraphs by moving from Lee’s 0.879n bound to a tighter fractional threshold: θ_STS^* ≤ 1 − x^* with x^* the unique root of p(x)=8x^3−22x^2+10x−1 in [0,1/6], numerically x^*≈0.1422, yielding θ_STS^* ≈0.8578. The authors adapt Delcourt–Postle’s fractional-decomposition framework, employing edge-gadgets and a sophisticated weighting w_H built from ordered 5-cliques to realize a perfect fractional Steiner triple system under a high essential codegree δ_2^{ess}(H)≥(1−x^*)v(H). A multi-stage optimization (P1→P5) shows that the resulting bound is tight for this approach, with the critical root x^* arising from a cubic, and the maximum weight achieved when e_0=1, leading to a univariate analysis. The results push toward the conjectured 3n/4 barrier, but the authors acknowledge that further improvement may require new ideas beyond the Delcourt–Postle scheme; they also discuss potential extensions to higher uniformities and related fractional-design questions.

Abstract

We establish an upper bound on the minimum codegree necessary for the existence of spanning, fractional Steiner triple systems in $3$-uniform hypergraphs. This improves upon a result by Lee in 2023. In particular, together with results from Lee's paper, our results imply that if $n$ is sufficiently large and satisfies some necessary divisibility conditions, then a $3$-uniform, $n$-vertex hypergraph $H$ contains a Steiner triple system if every pair of vertices forms an edge in $H$ with at least $0.8579n$ other vertices.

Codegree conditions for (fractional) Steiner triple systems

TL;DR

This work advances the threshold theory for Steiner triple systems in 3-uniform hypergraphs by moving from Lee’s 0.879n bound to a tighter fractional threshold: θ_STS^* ≤ 1 − x^* with x^* the unique root of p(x)=8x^3−22x^2+10x−1 in [0,1/6], numerically x^*≈0.1422, yielding θ_STS^* ≈0.8578. The authors adapt Delcourt–Postle’s fractional-decomposition framework, employing edge-gadgets and a sophisticated weighting w_H built from ordered 5-cliques to realize a perfect fractional Steiner triple system under a high essential codegree δ_2^{ess}(H)≥(1−x^*)v(H). A multi-stage optimization (P1→P5) shows that the resulting bound is tight for this approach, with the critical root x^* arising from a cubic, and the maximum weight achieved when e_0=1, leading to a univariate analysis. The results push toward the conjectured 3n/4 barrier, but the authors acknowledge that further improvement may require new ideas beyond the Delcourt–Postle scheme; they also discuss potential extensions to higher uniformities and related fractional-design questions.

Abstract

We establish an upper bound on the minimum codegree necessary for the existence of spanning, fractional Steiner triple systems in -uniform hypergraphs. This improves upon a result by Lee in 2023. In particular, together with results from Lee's paper, our results imply that if is sufficiently large and satisfies some necessary divisibility conditions, then a -uniform, -vertex hypergraph contains a Steiner triple system if every pair of vertices forms an edge in with at least other vertices.

Paper Structure

This paper contains 17 sections, 30 theorems, 116 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Fractional Steiner triple systems
  3. Related work
  4. Organization
  5. Edge-gadgets and the weighting
  6. Edge-gadgets
  7. The weighting
  8. Reformulation
  9. Optimization
  10. Reduction to 8 variables
  11. Reduction to two variables
  12. Reduction to one variable
  13. The final optimization
  14. Concluding remarks
  15. The conjectured value of $\theta_{\mathop{\mathrm{STS}}\nolimits}^\ast$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Every graph with $n \geq 3$ vertices and minimum degree at least $n / 2$ has a Hamiltonian cycle.

Figures (3)

  • Figure 1: Sketch for how the demand of $p$ is distributed among the $K_5^{(3)}$'s
  • Figure 2: Dependency diagram of the variables' constraints in (P3)
  • Figure 3: $\hat{W}_5(f)$ for $\textcolor{navyblue}{d=0}$ to $\textcolor{darkorange}{d=1/6}$ and $\textcolor{burgundy}{d=x^\ast}$

Theorems & Definitions (66)

  • Theorem 1.1: Dirac 1952, dirac_original
  • Corollary 1.2
  • Theorem 1.3: Lee 2023, hyunwoo
  • Remark 1.4
  • Theorem 1.5
  • Conjecture 1.6: Lee 2023, hyunwoo
  • Remark 1.7
  • Remark 1.8: Notation and terminology
  • Definition 1.9: $\theta_{\mathop{\mathrm{STS}}\nolimits}$
  • Definition 1.10: Weightings, perfect fractional Steiner triple systems
  • ...and 56 more