Let $\mathbb{Z}$ be the set of whole numbers and $l,m\in N$. Let's color all elements of $\mathbb{Z}\times\mathbb{Z}$ in $k$ different colors. Prove that we can find two aritmetic progressions $A$ and $B$ of length $l$ and $m$ in $\mathbb{Z}$ such that $A\times B$ contains only elements of the same color.

I know that this has something to do with Van der Waerden's theorem that has to be extended from $\mathbb{Z}^1$ to $\mathbb{Z}^2$ and I have found some pretty neat theorems here. But I could not make a direct connection.