I was asked the question
Prove that every $2$-colouring of the lattice points of $\mathbb R^m$ has a collection of $n$ monochromatic points whose centroid is a lattice point of the same colour
Now, I could prove it using Pigeon Hole Principle. The idea of that proof is
Consider $a_i$, $i\in \{1,2,\dots ,n\}$ to be $n$ red (WLOG) points all of whose coordinates are from $\{n\mathbb Z\}$ (i.e., multiples of $n$). Now, either $\frac 1n\sum a_i$ is red, or consider the $b_i$'s defined by $b_j=(n+1)a_j-\sum a_i$, where either a $b_k$ is red for some $k$, or all of them are blue including $\frac 1n\sum b_i=\frac 1n\sum a_i$.
But, I want to know if it is possible to prove this using Ramsey's Theorem. I was thinking about choosing $R(n+1,n+1)$ points to claim that we have a monochromatic $(n+1)$-clique. But, how can we take the centroid thing into consideration? This is what I can't figure out.
Edit: as said in the comments, Ramsey's Theorem doesn't give any structure for your set. So, it looks weird to try using only Ramsey Theorem. But, I got this question from the book on Graph Theory by Douglas B West, and at the end in brackets, the question had something like "try not using Ramsey Theorem, only PHP is enough". This made me assume that there must be a way to do it using Ramsey Theorem as well. Maybe it will also additionally need some PHP as well... I'm not sure