Paul Erdős and Eric Milner published in 1972 A theorem in the partition calculus, where they established that if is a countable ordinal and , then there is a countable ordinal such that
meaning that any graph whose set of vertices is either contains a clique (complete subgraph) whose set of vertices has order type or an independent set of size .
The result is false if is replaced by , except for when , in which case we can take as well, this is Ramsey’s theorem.
The least such that is , in which case a stronger result holds, namely . In fact, more is true: the homogeneous set of order type can be taken to be a stationary subset of , and the set of type can be required to be closed, meaning that its th member is the supremum of the other members of the set. Since stationary sets contain closed subsets of any countable order type, we see that holds for any countable ordinal , where the subindex cl indicates that the sets of vertices of type or are required to be closed on their supremum.
It is thus natural to wonder whether a closed version of the Erdős-Milner theorem holds. Jacob Hilton and I establish precisely this result in our paper Topological Ramsey numbers and countable ordinals.
This was a problem I had been curious about for a while, but kept not finding time to investigate. Finally I found a student at Boise State interested in working on this question for their master’s thesis, which gave me the perfect excuse to think seriously about it. I wrote a series of detailed notes for my student, who ended up leaving the program early, so I decided to continue and turn the notes into a paper. I even gave a preliminary talk on the results I had, together with some other results on the partition calculus of small countable ordinals. Hilton was a graduate student at that point, and he contacted me when he found out I was studying the problem, since this was precisely the topic of his dissertation. We decided to combine what we had, and soon we managed to extend our results and solve the full problem.
Many questions remain, as we believe the general bounds we found can be significantly improved, and it seems interesting to compute the optimal value of such that for specific values of and . Omer Mermelstein has some striking results in this direction.
Our paper appeared in Foundations of Mathematics, the proceedings of the conference in honor of Hugh Woodin’s 60th birthday. It can also be found on the arXiv and on my papers page.