Erdős Problem 532

References:

• erdosproblems.com/532
• [Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (1973), 117-138.
• [Er75b] Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (1975), 295-310.
• [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (1977), 43-72.
• [Hi74] Hindman, Neil, Finite sums from sequences within cells of a partition of $\mathbb{N}$. J. Combinatorial Theory Ser. A (1974), 1-11.

theorem Erdos532.erdos_532 : True ↔ ∀ (c : ℕ → Fin 2), ∃ (A : Set ℕ), A.Infinite ∧ ∃ (color : Fin 2), ∀ (S : Finset ℕ), S.Nonempty → ↑S ⊆ A → c (∑ n ∈ S, n) = color

If $\mathbb{N}$ is 2-coloured then is there some infinite set $A\subseteq \mathbb{N}$ such that all finite subset sums[ \sum_{n\in S}n](as $S$ ranges over all non-empty finite subsets of $A$) are monochromatic?

Asked by Graham and Rothschild. Proved by Hindman [Hi74] (for any number of colours).