Influential coalitions for boolean functions I: Constructions

For f : {0, 1}ⁿ → {0, 1} and S ⊂ {1, 2, . . .,,n}, let J⁺ _S(f) be the probability that, for x uniform from {0. 1}ⁿ, there is some y ϵ {0, 1}ⁿ with f(y) = 1 and x = y outside S. We are interested in estimating, for given μ(f) (:= E(f)) and m, the least possible value of max {J⁺ _s(f): |S| = m}. A theorem of Kahn, Kalai, and Linial (KKL) gave some understanding of this issue and led to several stronger conjectures. Here we disprove a pair of conjectures from the late 80s, as follows.