On strong infinite Sidon and Bh sets and random sets of integers

© 2021 ElsevierA set of integers S ¿ N is an a–strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on a, more specifically if (x + w) - (y + z) = max{xa, ya, za, wa} for every x, y, z, w ¿ S satisfying max{x, w} 6= max{y, z}. We obtain a new lower bound for the growth of a–strong infinite Sidon sets when 0 = a < 1. We also further extend that notion in a natural way by obtaining the first non-trivial bound for a–strong infinite Bh sets. In both cases, we study the implications of these bounds for the density of, respectively, the largest Sidon or Bh set contained in a random infinite subset of N. Our theorems improve on previous results by Kohayakawa, Lee, Moreira and Rödl.Peer ReviewedPostprint (author's final draft