Worst-case interactive communication - II: Two messages are not optimal

X and Y are random variables. Person P X knows X, Person P Y knows Y , and both know the joint probability distribution of the pair (X; Y). Using a predetermined protocol, they communicate over a binary, error-free, channel in order for P Y to learn X. P X may or may not learn Y. How many information bits must be transmitted (by both persons) in the worst case if only m messages are allowed? ^ C 1 (XjY) is the number of bits required when at most one message is allowed, necessarily from P X to P Y. ^ C 2 (XjY) is the number of bits required when at most two messages are permitted: P Y transmits a message to P X , then P X responds with a message to P Y. ^ C 1 (XjY) is the number of bits required when communication is unrestricted: P X and P Y can communicate back and forth. It is known that one-message communication may require exponentially more bits than the minimum necessary: for some (X; Y) pairs, ^ C 1 (XjY) = 2 ^ C1(XjY)1. Yet just two messages suuce to reduce communication to almost the minimum: for all (X; Y) pairs ^ C 2 (XjY) 4 ^ C 1 (XjY) + 3. It was further shown that for a large class of (X; Y) pairs, two messages are optimal: ^ C 2 (XjY) = ^ C 1 (XjY). It remained uncertain whether two message are optimal for all (X; Y) pairs. In this paper we deene the chromatic-decomposition number of a hypergraph and show that under general conditions on (X; Y) it can be used to determine ^ C 2 (XjY). We use this result to prove that for some (X; Y) pairs, two-message communication requires twice the minimum number of bits: for all positive and c we present an (X; Y) pair for which ^ C 2 (XjY) (2) ^ C 1 (XjY) c.