All Random Solved Random Open
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?
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).