Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Independent Sets in Hypergraphs

Jacques Verstraete, Chase Wilson

TL;DR

The paper extends Shearer's method for triangle-free graphs to (r+1)-uniform locally sparse hypergraphs, establishing a lower bound on the independence number of the form $\alpha(H) \ge c_r (\log d/d)^{1/r} \cdot n$ with explicit constants. By introducing a non-uniform sampling of independent sets weighted by $\exp(-\delta|\partial_r I|)$ and analyzing a per-vertex statistic $X_v$ (assembled into $X$), the authors derive complementary upper and lower bounds on $\mathbb{E}(X)$ that yield the desired bound. The approach yields improved constants for small $r$ (e.g., $c_2 = 1/8$ for large $d$) and provides a short, self-contained proof of the Ajtai–Komlós–Pintz–Spencer–Szemerédi result for a broader class of locally sparse hypergraphs. The work also motivates conjectures on optimal constants and the role of higher-shadow terms $\partial_k I$ in shaping independence number bounds.

Abstract

A theorem of Shearer states that every $n$-vertex triangle-free graph of maximum degree $d \geq 2$ contains an independent set of size at least $(d\log d - d + 1)/(d - 1)^2 \cdot n$. Ajtai, Komlós, Pintz, Spencer and Szemerédi proved that every $(r + 1)$-uniform $n$-vertex ``uncrowded'' hypergraph of maximum degree $d \geq 1$ has an independent set of size at least $c_r(\log d)^{1/r}/d^{1/r} \cdot n$ for some $c_r > 0$ depending only on $r$. Shearer asked whether his method for triangle-free graphs could be extended to uniform hypergraphs. In this paper, we answer this in the affirmative, thereby giving a short proof of the theorem of Ajtai, Komlós, Pintz, Spencer and Szemerédi for a wider class of ``locally sparse'' hypergraphs.

Independent Sets in Hypergraphs

TL;DR

The paper extends Shearer's method for triangle-free graphs to (r+1)-uniform locally sparse hypergraphs, establishing a lower bound on the independence number of the form with explicit constants. By introducing a non-uniform sampling of independent sets weighted by and analyzing a per-vertex statistic (assembled into ), the authors derive complementary upper and lower bounds on that yield the desired bound. The approach yields improved constants for small (e.g., for large ) and provides a short, self-contained proof of the Ajtai–Komlós–Pintz–Spencer–Szemerédi result for a broader class of locally sparse hypergraphs. The work also motivates conjectures on optimal constants and the role of higher-shadow terms in shaping independence number bounds.

Abstract

A theorem of Shearer states that every -vertex triangle-free graph of maximum degree contains an independent set of size at least . Ajtai, Komlós, Pintz, Spencer and Szemerédi proved that every -uniform -vertex ``uncrowded'' hypergraph of maximum degree has an independent set of size at least for some depending only on . Shearer asked whether his method for triangle-free graphs could be extended to uniform hypergraphs. In this paper, we answer this in the affirmative, thereby giving a short proof of the theorem of Ajtai, Komlós, Pintz, Spencer and Szemerédi for a wider class of ``locally sparse'' hypergraphs.
Paper Structure (8 sections, 1 theorem, 49 equations)

This paper contains 8 sections, 1 theorem, 49 equations.

Table of Contents

  1. Introduction
  2. A Non-Uniform Distribution on Independent Sets
  3. Proof of Theorem \ref{['main']}
  4. Definition of parameters
  5. The upper bound on $\mathrm{E}(X)$
  6. The lower bound on $\mathrm{E}(X)$
  7. Proof of Theorem \ref{['main']}
  8. Concluding remarks

Key Result

Theorem 1

For $r \geq 1$, there exists $c_r > 0$ such that for any $(r + 1)$-uniform $n$-vertex locally sparse hypergraph $H$ with maximum degree $d \geq 1$. where $c_r = (1 - o_d(1)) f(r)$ for some function $f$.

Theorems & Definitions (4)

  • Theorem 1
  • Conjecture A
  • Conjecture B
  • Conjecture C