An application of graph theory to additive number theory (Q1068128)

It is proved that, if \({\mathfrak A}=a_1<a_2<...<a_n\) is a sequence of positive integers such that no integer can be expressed as a sum \(a_i+a_j\) in more than k ways, then \({\mathfrak A}\) is the union of \(C_1(k) n^{1/3}\) \(B_2\)-sequences, a \(B_2\)-sequence being a sequence with all two-element sums distinct. On the other hand, such an \({\mathfrak A}\) exists which is not the union of \(C_2(k) n^{1/3}\) \(B_2\)- sequences. Proofs are couched in terms of hypergraphs.