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Partial results for union-closed conjectures on the weighted cube

Gabriel Gendler

TL;DR

The paper studies the union-closed conjecture on the $\vec{p}$-weighted Boolean cube with product Bernoulli measure $\mu_{\vec{p}}$. It extends Karpas' density-1/2 result to this weighted setting using a Fourier-analytic framework based on influences. It also generalizes Knill's logarithmic lower bound to the weighted cube, providing quantitative relations between $\mu(\mathcal{F})$ and $\mu(\mathcal{F}_i)$. Together, these results advance understanding of union-closed families under nonuniform product measures and connect Fourier analysis with a classical combinatorial conjecture.

Abstract

The celebrated union-closed conjecture is concerned with the cardinalities of various subsets of the Boolean $d$-cube. The cardinality of such a set is equivalent, up to a constant, to its measure under the uniform distribution, so we can pose more general conjectures by choosing a different probability distribution on the cube. In particular, for any sequence of probabilities $(p_i)_{i=1}^d$ we can consider the product of $d$ independent Bernoulli random variables, with success probabilities $p_i$. In this short note, we find a generalised form of Karpas' special case of the union-closed conjecture for families $\mathcal{F}$ with density at least half. We also generalise Knill's logarithmic lower bound.

Partial results for union-closed conjectures on the weighted cube

TL;DR

The paper studies the union-closed conjecture on the -weighted Boolean cube with product Bernoulli measure . It extends Karpas' density-1/2 result to this weighted setting using a Fourier-analytic framework based on influences. It also generalizes Knill's logarithmic lower bound to the weighted cube, providing quantitative relations between and . Together, these results advance understanding of union-closed families under nonuniform product measures and connect Fourier analysis with a classical combinatorial conjecture.

Abstract

The celebrated union-closed conjecture is concerned with the cardinalities of various subsets of the Boolean -cube. The cardinality of such a set is equivalent, up to a constant, to its measure under the uniform distribution, so we can pose more general conjectures by choosing a different probability distribution on the cube. In particular, for any sequence of probabilities we can consider the product of independent Bernoulli random variables, with success probabilities . In this short note, we find a generalised form of Karpas' special case of the union-closed conjecture for families with density at least half. We also generalise Knill's logarithmic lower bound.

Paper Structure

This paper contains 4 sections, 10 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Families with large weight
  3. A logarithmic bound for the weighted cube
  4. Acknowledgements

Key Result

Theorem 1.2

Let $\mathcal{F}$ be a union-closed family in $\mathcal{P}([d])$ with $|\mathcal{F}| \geq 2^{d-1}$. Then there exists $i \in [d]$ such that $|\mathcal{F}_i|/|\mathcal{F}| \geq 1/2$.

Theorems & Definitions (18)

  • Conjecture 1.1
  • Theorem 1.2: Karpas
  • Theorem 1.3: Knill
  • Definition 2.1: Influence
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Theorem 2.5
  • proof
  • ...and 8 more