The number of B 3 -sets of a given cardinality

A set S of integers is a B 3 -set if all the sums of the form a 1 + a 2 + a 3 , with a 1 , a 2 and a 3 ? S and a 1 ? a 2 ? a 3 , are distinct. We obtain asymptotic bounds for the number of B 3 -sets of a given cardinality contained in the interval n = { 1 , ? , n } . We use these results to estimate the maximum size of a B 3 -set contained in a typical (random) subset of n of a given cardinality. These results confirm conjectures recently put forward by the authors On the number of B h -sets, Combin. Probab. Comput. 25 (2016), no. 1, 108-127.