A Bound on Partitioning Clusters.

Let X be a finite collection of sets (or "clusters"). We consider the problem of counting the number of ways a cluster A is an element of X can be partitioned into two disjoint clusters A(1), A(2) is an element of X, thus A = A(1) (sic) A(2) is the disjoint union of A(1) and A(2); this problem arises in the run time analysis of the ASTRAL algorithm in phylogenetic reconstruction. We obtain the bound vertical bar{(A(1), A(2), A) is an element of X x X x X : A - A(1) (sic) A(2)}vertical bar <= vertical bar X vertical bar(3/p) where vertical bar X vertical bar denotes the cardinality of X, and p := log(3) 27/4 = 1.73814 ..., so that 3/p= 1.72598 .... Furthermore, the exponent p cannot be replaced by any largerquantity. This improves upon the trivial bound of vertical bar X vertical bar(2) The argument relies onestablishing a one-dimensional convolution inequality that can be established byelementary calculus combined with some numerical verification. In a similar vein, we show that for any subset A of a discrete cube {0, 1}(n), theadditive energy of A (the number of quadruples (a(1), a(2), a(3), a(4)) in A(4) with a(1) + a(2) =a(3) + a(4)) is at most vertical bar A vertical bar(log2) (6), and that this exponent is best possible.