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A short proof of Kahn-Kalai conjecture

P. Tran, V. Vu

TL;DR

The paper presents a concise, induction-based proof of the Kahn–Kalai conjecture and its $\varepsilon$-version by establishing a covering theorem for $l$-bounded increasing families. The approach hinges on a cost function $f_p(F)$ and a double-counting lemma, enabling an induction on $l$ and $N$ to show that $\langle F\rangle$ captures a fixed fraction of the $m_l$-level, with $m_l$ growing like $LpN\log(l+1)$. By optimizing constants, the authors reduce the effective universal constant $K$ to approximately $3.998$ in the $l\to\infty$ regime and extend the result to arbitrary small $\varepsilon_1$ via an additional $96 pN \log(1/\varepsilon_1)$ term. This work yields a shorter, accessible argument that sharpens the quantitative bounds in the threshold phenomena described by the Park–Pham result and broadens applicability to related random-structure thresholds.

Abstract

In a recent paper, Park and Pham famously proved Kahn-Kalai conjecture. In this note, we simplify their proof, using an induction to replace the original analysis. This reduces the proof to one page, and from the argument it is also easy to read that one can set the constant $K$ in the conjecture to $\approx 3.998$, which could be the best value under the current method. Our argument also applies to the $ε$-version of Park-Pham result, studied by Bell.

A short proof of Kahn-Kalai conjecture

TL;DR

The paper presents a concise, induction-based proof of the Kahn–Kalai conjecture and its -version by establishing a covering theorem for -bounded increasing families. The approach hinges on a cost function and a double-counting lemma, enabling an induction on and to show that captures a fixed fraction of the -level, with growing like . By optimizing constants, the authors reduce the effective universal constant to approximately in the regime and extend the result to arbitrary small via an additional term. This work yields a shorter, accessible argument that sharpens the quantitative bounds in the threshold phenomena described by the Park–Pham result and broadens applicability to related random-structure thresholds.

Abstract

In a recent paper, Park and Pham famously proved Kahn-Kalai conjecture. In this note, we simplify their proof, using an induction to replace the original analysis. This reduces the proof to one page, and from the argument it is also easy to read that one can set the constant in the conjecture to , which could be the best value under the current method. Our argument also applies to the -version of Park-Pham result, studied by Bell.
Paper Structure (4 sections, 6 theorems, 13 equations)

This paper contains 4 sections, 6 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. The covering theorem
  3. Reducing the constant $K$
  4. Covering theorem for arbitrary small $\epsilon_1$

Key Result

Theorem 1.1

There is a constant $K >0$ such that for any $l$-bounded increasing family $F,$$p_c (F) \le K q (F) \log (l+1).$

Theorems & Definitions (11)

  • Theorem 1.1: Park-Pham PP
  • Theorem 2.1: Park-Pham PP
  • Theorem 2.2: Covering theorem
  • proof
  • Lemma 2.1: Double counting lemma
  • proof
  • Remark 2.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • ...and 1 more