On strong infinite Sidon and $B_h$ sets and random sets of integers

A set of integers $S \subset \mathbb{N}$ is an $\alpha$-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on $\alpha$, more specifically if $| (x+w) - (y+z) | \geq \max \{ x^{\alpha},y^{\alpha},z^{\alpha},w^\alpha \}$ for every $x,y,z,w \in S$ satisfying $\max \{x,w\} \neq \max \{y,z\}$. We obtain a new lower bound for the growth of $\alpha$-strong infinite Sidon sets when $0 \leq \alpha < 1$. We also further extend that notion in a natural way by obtaining the first non-trivial bound for $\alpha$-strong infinite $B_h$ sets. In both cases, we study the implications of these bounds for the density of, respectively, the largest Sidon or $B_h$ set contained in a random infinite subset of $\mathbb{N}$. Our theorems improve on previous results by Kohayakawa, Lee, Moreira and R\"odl.