The cardinality of a set containing the pairwise sums of four positive integers

Choi, Erdős and Szemerédi showed that there exists an absolute constant $C$ such that for all subsets $A \subseteq \{1, 2, \ldots, 2n\}$ with at least $n+C$ elements, there exist four distinct positive integers whose pairwise sums are all contained in $A$. A proof that one can take $C = 3166$ was re