  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.
