The number of B3-sets of a given cardinality

A set S   of integers is a B3B3-set   if all the sums of the form a1+a2+a3a1+a2+a3, with a1a1, a2a2 and a3∈Sa3∈S and a1≤a2≤a3a1≤a2≤a3, are distinct. We obtain asymptotic bounds for the number of B3B3-sets of a given cardinality contained in the interval [n]={1,…,n}[n]={1,…,n}. We use these results to estimate the maximum size of a B3B3-set contained in a typical (random) subset of [n][n] of a given cardinality. These results confirm conjectures recently put forward by the authors [On the number of  BhBh-sets, Combin. Probab. Comput. 25 (2016), no. 1, 108–127].