Erdős Problem #656

Let \(A\subset \mathbb{N}\) be a set with positive upper density. Must there exist an infinite set \(B\subseteq A\) and integer \(t\) such that \[\{b_1+b_2: b_1\neq b_2\in B\}+t\subseteq A?\] Erdős [Er75b] posed this as a candidate for a density version of Hindman's theorem. This is true, and was proved by Kra, Moreira, Richter, and Robertson [KMRR24, Theorem 1.2 (i)], even for sets of positive upper Banach density and with \(t \geq 0\). The inclusion says that every \(b_1+b_2+t\) is an element of `A`, as an integer.