Entropy bound for pairwise-union tuples
= Entropy bound for pairwise-union tuples
Let $A_1,\ldots,A_m$ be random subsets of a common ground set. If every pairwise union $A_i\cup A_j$ takes values in a fixed family $\mathcal A$ of sets of cardinality at most $k$, then <Shearer's inequality> and the fact that a pair of bits with prescribed OR has at most three possibilities give
$$
(m-1)H(A_1,\ldots,A_m)
\leq\sum_{i<j}\bigl(\log|\mathcal A|+k\log3\bigr).
$$