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Fractional Vs. Expectation Thresholds: Random Support Case

Thomas Fischer, Yury Person

TL;DR

This paper addresses Talagrand's conjecture on the relationship between fractional and ordinary expectation thresholds in random discrete structures by focusing on the average-case, unweighted setting where the threshold support is a random $k$-uniform hypergraph. The authors formalize weight and upset notions, and prove that for $g= rac{1}{r}oldsymbol{1}_{H^{(k)}(n,m)}$ (and for $H^{(k)}(n,q)$) with $L\, ext{ge}\,4e^{3}$, the condition $w(g,p)=1$ implies the existence of a cover $G$ with $ig<gig>\subseteqig<Gig>$ and $wig(G, rac{p}{L}ig)\le 1$ a.a.s., thereby establishing a constant-factor control of the thresholds in this random setting. The proof combines a constructive covering strategy—partitioning into $G_0$ and a family of $G_j$—with delicate probabilistic bounds (using hypergeometric tails and negative correlations) to show the weighted sum remains below the target. The results strengthen the understanding of threshold relations in random hypergraphs and point toward broader average-case validations of Talagrand’s conjecture, with potential extensions to weighted or pseudorandom frameworks. ∎

Abstract

A conjecture of Talagrand (2010) states that the so-called expectation and fractional expectation thresholds are always within at most some constant factor from each other. We prove for the unweighted case that this is a.a.s. true when the support is a random hypergraph.

Fractional Vs. Expectation Thresholds: Random Support Case

TL;DR

This paper addresses Talagrand's conjecture on the relationship between fractional and ordinary expectation thresholds in random discrete structures by focusing on the average-case, unweighted setting where the threshold support is a random -uniform hypergraph. The authors formalize weight and upset notions, and prove that for (and for ) with , the condition implies the existence of a cover with and a.a.s., thereby establishing a constant-factor control of the thresholds in this random setting. The proof combines a constructive covering strategy—partitioning into and a family of —with delicate probabilistic bounds (using hypergeometric tails and negative correlations) to show the weighted sum remains below the target. The results strengthen the understanding of threshold relations in random hypergraphs and point toward broader average-case validations of Talagrand’s conjecture, with potential extensions to weighted or pseudorandom frameworks. ∎

Abstract

A conjecture of Talagrand (2010) states that the so-called expectation and fractional expectation thresholds are always within at most some constant factor from each other. We prove for the unweighted case that this is a.a.s. true when the support is a random hypergraph.
Paper Structure (4 sections, 4 theorems, 30 equations)

This paper contains 4 sections, 4 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Helpful Estimates
  3. Proof of Theorem \ref{['theorem:main']}
  4. Concluding Remarks

Key Result

Theorem 2

Let $k$, $r\in\mathbb{N}$ and set $g:= \frac{1}{r}\cdot\mathds{1}_{H^{(k)}(n,m)}$. Then for $L\geq 4\cdot e^3$ the following holds a.a.s.We are considering probabilities of events which tend to $1$ as $n$ goes to infinity. If $p\in[0,1]$ is such that $w(g,p)=1$ then there exists $G\subseteq 2^X\setm

Theorems & Definitions (24)

  • Conjecture 1: Conjecture 6 from FP23
  • Theorem 2
  • Corollary 3
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Claim 7
  • proof
  • ...and 14 more