OPEN
This is open, and cannot be resolved with a finite computation.
Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?
First asked by Hindman. Hindman
[Hi80] has proved this is false (with 7 colours) if we ask for an infinite $A$. In
[Er77c] Erdős asks about the case for an infinite $A$ with just $2$ colours (see
[1198]).
Moreira
[Mo17] has proved that in any finite colouring of $\mathbb{N}$ there exist $x,y$ such that $\{x,x+y,xy\}$ are all the same colour.
Alweiss
[Al23] has proved that in any finite colouring of $\mathbb{Q}\backslash \{0\}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour. Bowen and Sabok
[BoSa22] had proved this earlier for the first non-trivial case of $\lvert A\rvert=2$.
Additional thanks to: Ryan Alweiss
