Erdős Problem #1198

If \(\mathbb{N}\) is \(2\)-coloured, must there exist an infinite set \(A=\{a_1<a_2<\cdots\}\) such that all expressions of the shape \[\sum_{j=1}^{k}\prod_{i\in S_j}a_i,\] for \(k\geq 1\) and pairwise disjoint nonempty finite sets of indices \(S_1,\ldots,S_k\) (excluding the trivial expressions \(a_i\), that is \(k=1\) and \(|S_1|=1\)), have the same colour? Erdős [Er80, p.104] wrote that "one would perhaps guess that the answer must be 'no' but no counterexample is in sight". The case where every \(S_j\) is a singleton is Hindman's theorem [Hi74]. The answer is no: a counterexample was given by Smith [Sm95].