Problem 866

Erdős Problem #866: declared status 'open'. Formalized: no. Let $k\geq 3$ and $g_k(N)$ be minimal such that if $A\subseteq \{1,\ldots,2N\}$ has $\lvert A\rvert \geq N+g_k(N)$ then there exist integers $b_1,\ldots,b_k$ such that all $\binom{k}{2}$ pairwise sums are in $A$ (but the $b_i$ themselves need not be in $A$). Estimate $g_k(N)$. Prize: no. Tags: additive combinatorics, number theory.