On the FKB conjecture for disjoint intersections

Abstract

We reformulate an outstanding combinatorial conjecture of van den Berg and Kesten which, if true, would have many nice consequences in percolation theory. Let Ω n={0, 1} n with x∈Ω n , K⊆{1,…, n} and K c ={1,…, n}\\ K. Define c( x, K) as the set of y∈Ω n for which y j = x j for all j∈ K. Our reformulation is that for all n⩾1 and 1⩽ m⩽2 n , all distinct x 1,…, x m∈Ω n and all K 1,…, K m ⊆{1,…, n}, m2 n⩽ ∪ i=1 m c(x i,K i · ∪ i=1 m c(x i,K c i) . This is dual to a version of the van den Berg–Kesten conjecture examined by van den Berg and Fiebig that we refer to as the FKB conjecture. It avoids direct use of their disjoint intersection operation, and we have found it easier to work with. The equivalence of (∗) and the FKB conjecture is easily proved, and we verify cases of (∗) which are new and of course settle special cases of the FKB conjecture. Our original aim was to verify (∗) by induction on m and n, and although we have not done this our partial induction results offer hope that the van den Berg–Kesten conjecture will eventually be established in this way. Those results and a few other facts yield confirmation of (∗) for m⩽6 and, with the exception of n=4, for m=7,8.