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On the "second" Kahn--Kalai Conjecture: cliques, cycles, and trees

Quentin Dubroff, Jeff Kahn, Jinyoung Park

TL;DR

The paper advances the second Kahn–Kalai Conjecture by establishing $N(H,F)<\mathbb{E}_p X_F$ for three simple graph families—cliques, cycles, and trees with bounded maximum degree—when $p= L q$ and $H$ is $q$-sparse. It develops a two-pronged tree-analysis, combining a low-$D(\underline d)$ bound and a high-$D(\underline d)$ bound via a legal-degree-sequence decomposition, to show the desired inequality for trees with $L$ depending only on $\Delta$. For cliques and cycles, it provides regime-based arguments: small-$q$ counting bounds via edge-sharing structures, and large-$q$ density-contradiction or path-extension techniques to force the desired bound. The methods integrate edge-disjoint decompositions, hypergraph degree arguments, and path-based cycle analysis, offering a concrete framework toward the general conjecture and highlighting the distinct challenges in these elementary cases.

Abstract

We prove a few simple cases of a random graph statement that would imply the "second" Kahn--Kalai Conjecture. Even these cases turn out to be reasonably challenging, and it is hoped that the ideas introduced here may lead to further interest in, and further progress on, this natural problem.

On the "second" Kahn--Kalai Conjecture: cliques, cycles, and trees

TL;DR

The paper advances the second Kahn–Kalai Conjecture by establishing for three simple graph families—cliques, cycles, and trees with bounded maximum degree—when and is -sparse. It develops a two-pronged tree-analysis, combining a low- bound and a high- bound via a legal-degree-sequence decomposition, to show the desired inequality for trees with depending only on . For cliques and cycles, it provides regime-based arguments: small- counting bounds via edge-sharing structures, and large- density-contradiction or path-extension techniques to force the desired bound. The methods integrate edge-disjoint decompositions, hypergraph degree arguments, and path-based cycle analysis, offering a concrete framework toward the general conjecture and highlighting the distinct challenges in these elementary cases.

Abstract

We prove a few simple cases of a random graph statement that would imply the "second" Kahn--Kalai Conjecture. Even these cases turn out to be reasonably challenging, and it is hoped that the ideas introduced here may lead to further interest in, and further progress on, this natural problem.

Paper Structure

This paper contains 8 sections, 21 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cliques
  4. Trees
  5. Set-up and definitions.
  6. First bound.
  7. Second bound.
  8. Cycles

Key Result

Theorem 1.3

dubroff2025second There is a fixed $K$ such that if $H$ is $q$-sparse and $p=Kq\log^2 n$, then Furthermore, there is a fixed $\alpha >0$ such that eq.MT holds if $H$ is $q$-sparse with $q=\alpha p\le 1/(3n)$.

Theorems & Definitions (53)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • ...and 43 more