A BK inequality for randomly drawn subsets of fixed size
  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. 
J. van den Berg and J. Jonasson
math.PR