We don't include the proof of this general statement here, but the more ambitious students may attempt it on their own. Note that the case $s=1$ is just the Pigeon Hole Principle, while the case $s=r=2$ is just Ramsey's Theorem for Graphs. An argument using double induction is required for the proof in the general case. The first induction is on $r$ and the second is on $s\text{.}$