Problem Statement
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?
Categories:
Number Theory Ramsey Theory
Progress
In other words, must some colour class be an IP set. Asked by Graham and Rothschild. See also [531].Proved by Hindman [Hi74] (for any number of colours).
See also [948].
Source: erdosproblems.com/532 | Last verified: January 15, 2026