Assume that A, B are finite families of n-element sets.
We prove that there is an element that simultaneously
belongs to at least |A|/2n sets
in A and to at least |B|/2n sets in B. We use this result to prove
that for any inconsistent DNF's F,G with OR of fanin m and ANDs
of fanin n there is a decision tree of depth
(4n ln m + 2) separating F from G.