Erdős Problem #772 — Large Sidon subsets of bounded-representation sets

For fixed $k≥1$, let $H_k(n)$ be the largest integer such that every $n$-element set of integers in which each integer has at most $k$ representations as a sum of two elements contains a Sidon subset of size $H_k(n)$. Does $H_k(n)/√n→∞$? Is $H_k(n)>n^{1/2+c}$ for some absolute $c>0$?