Colour in the grid $\mathbb{Z}\times\mathbb{Z} = \{(i,j): i,j \in\mathbb{Z}\}$ using $R$ colours.
Use Ramsey's Theorem to prove the following:
For each $K\geq 1$ there is a monochromatic $K\times K$ subgrid $\{(i,j):i\in I, j\in J\}$ of $I,J\subseteq \mathbb{Z}$ and $|I| = |J| = K$ (so the subgrid has not necessarily consecutive columns or rows).
How would I start this? I'm having trouble visualising or drawing smaller examples. Is this a famous result?
To see a connection to Ramsey's theorem, there is a well-known correspondence between the cells of an $n \times m$ grid and the edges of the complete bipartite graph $K_{n,m}$ with $n$ vertices on one side and $m$ on the other side of the bipartition.
If you let $\{x_1,\dots,x_n\}$ be one half of the bipartition and $\{y_1, \dots, y_m\}$ be the other half, then the cell in position $(i,j)$ corresponds to the edge $x_i y_j$.
So, often you will see an arbitrary bipartite graph being represented by a "biadjacency matrix" in which the entry $(i,j)$ is $1$ or $0$ according to whether the edge $x_i y_j$ is present or absent. Or, as here, a coloring of the $n \times m$ grid corresponds to an edge-coloring of $K_{n,m}$.
Under this bijection, and the usual replacement of "infinite" by "sufficiently large", your desired statement becomes a statement about graph theory:
For all $k$ and $r$, there is a sufficiently large $n$ such that in any $r$-color edge coloring of $K_{n,n}$ we will be able to find a monochromatic copy of $K_{k,k}$.
I don't offhand see a way to immediately get this as a corollary of Ramsey's theorem for coloring $K_n$, but you should be able to easily adapt the same proof technique to get an upper bound of $k \cdot r^{(r-1)k+1}$ on the least value of $n$ needed.
Of course, the same proof technique could be applied to the grid directly.
