11.5: Ramsey's Theorem
By this time, you are probably not surprised to see that there is a very general form of Ramsey's theorem. We have a bounded number of bins or colors and we are placing the subsets of a fixed size into these categories. The conclusion is that there is a large set which is treated uniformly.
Here's the formal statement.
Let \(r\) and \(s\) be positive integers and let \(\textbf{h}=(h_1,h_2,…,h_r)\) be a string of integers with \(h_i \geq s\) for each \(i=1,2,…,s\). Then there exists a least positive integer \(R(s:h_1,h_2,…,h_r)\) so that if \(n \geq n_0\) and \(\phi:C([n],s] \rightarrow [r]\) is any function, then there exists an integer \(\alpha \in [r]\) and a subset \(H_{\alpha}⊆[n]\) with \(|H_{\alpha}|=h_{\alpha}\) so that \(\phi (S)= \alpha\) for every \(S \in C(H_{\alpha},s)\).
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 \(s\).