On the ubiquity of Sidon sets

A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B_2[g]-set is a set A of integers such that no integer has more than g essentially distinct representations as the sum of two elements of A. It is proved that almost all small sumsets of {1,2,…,n} are B_2[g]-sets, in the sense that if B_2g denotes the number of B_2[g]-sets of cardinality k contained in the interval {1,2,…,n}, then lim_{n\to\infty} B_2g/\binom{n}{k} = 1 if k = o(n^{g/(2g+2)}).