A BK inequality for randomly drawn subsets of fixed size

TL;DR

This paper proves a BK inequality for measures that select subsets of fixed size uniformly, extending the inequality's applicability beyond product measures to these specific non-product measures.

Contribution

The paper establishes the BK inequality for fixed-size subset measures, a natural non-product measure, and extends the result to weighted and product measures.

Findings

Proved BK inequality for uniform fixed-size subset measures.

Extended the inequality to weighted versions of these measures.

Demonstrated applicability to products of such measures.

Abstract

The BK inequality (\cite{BK85}) says that,for product measures on $\{0,1\}^n$, the probability that two increasing events $A$ and $B$ `occur disjointly' is at most the product of the two individual probabilities. The conjecture in \cite{BK85} that this holds for {\em all} events was proved by Reimer (cite{R00}). Several other problems in this area remained open. For instance, although it is easy to see that non-product measures cannot satisfy the above inequality for {\em all} events,there are several such measures which, intuitively, should satisfy the inequality for all{\em increasing} events. One of the most natural candidates is the measure assigning equal probabilities to all configurations with exactly $k$ 1's (and probability 0 to all other configurations). The main contribution of this paper is a proof for these measures. We also point out how our result extends to weighted versions of these measures, and to products of such measures.