A note on positive association

TL;DR

This paper proves a new independence property for increasing subsets of the hypercube under product measures, addressing a question posed by J. Steif and relating to S. Sahi's conjecture.

Contribution

It establishes a novel independence result for increasing sets, providing insights into longstanding open problems in probability and combinatorics.

Findings

Shows that specific pairwise independence implies overall independence for increasing sets

Answers a question posed by J. Steif

Connects to a conjecture of S. Sahi

Abstract

We show that if ${\mathcal A},{\mathcal B},{\mathcal C}$ are increasing subsets of $\Omega:=\{0,1\}^n$ with ${\mathcal A}\neq\emptyset$, then with respect to any product probability measure on $\Omega$, \[ \mbox{if each of the pairs $\{{\mathcal A}\cap{\mathcal B},{\mathcal C}\}$, $\{{\mathcal A}\cap {\mathcal C},{\mathcal A}\}$ is independent, then ${\mathcal B}$ and ${\mathcal C}$ are independent.} \] This implies an answer to a motivating question of J. Steif, and is related to a basic, still open variant of that question, and to a well-known conjecture of S. Sahi.