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Let x 1 , . . . , x k be n-bit numbers and T ∈ N. Assume that P 1 , . . . , P k are players such that P i knows all of the numbers except x i . The players want to determine if k j=1 x j = T by broadcasting as few bits as possible. Chandra, Furst, and Lipton obtained an upper bound of O( √ n) bits for the k = 3 case, and a lower bound of ω(1) for k ≥ 3 when T = Θ(2 n ). We obtain (1) for general k ≥ 3 an upper bound of k + O(n 1/(k−1) ), ( 2) for k = 3, T = Θ(2 n ), a lower bound of Ω(log log n), (3) a generalization of the protocol to abelian groups, (4) lower bounds on the multiparty communication complexity of some regular languages, (5) lower bounds on branching programs, and (6) empirical results for the k = 3 case.

There has been much work on the following question: given n, how large can a subset of {1, . . . , n} be that has no arithmetic progressions of length 3. We call such sets 3-free. Most of the work has been asymptotic. In this paper we sketch applications of large 3-free sets, present techniques to find large 3-free sets of {1, . . . , n} for n 250, and give empirical results obtained by coding up those techniques. In the sequel we survey the known techniques for finding large 3-free sets of {1, . . . , n} for large n, discuss variants of them, and give empirical results obtained by coding up those techniques and variants.