I was an undergraduate student at Los Andes, from 1992 to 1996. This year, their mathematics program is turning 50. There was a conference in September to celebrate the event, and I had the honor to give one of the talks (see here for the English version of the slides).
The topic is the partition calculus of very small countable ordinals (mainly ordinals below , actually). The paper reviews Ramsey’s theorem and a few finite examples, before discussing the two main results.
One is an old theorem by Haddad and Sabbagh, unfortunately not well known. In 1969, they computed the Ramsey numbers for finite.
Given nonzero ordinals , recall that is the least such that any red-blue coloring of the edges of either admits a red copy of or a blue copy of . Clearly , if , and , so we may as well assume that , and we adopt this convention in what follows.
Ramsey proved two theorems about this function in a famous 1928 paper that introduced the topic. In the notation we have just set up, his first result asserts that is finite whenever are finite, and his second result states that . The computation of the numbers is an extremely difficult, most likely unfeasible, problem, though is obviously a recursive function. We are concerned here with the values of the function when at least one of the arguments is infinite.
It turns out that is already . Hence, if we are interested in studying the countable values of the function , then we must assume that either , in which case and there is nothing more to say, or else (that is, if is countable and strictly larger than ) we must assume that is finite.
The function has been intensively studied when is a limit ordinal, particularly a power of . Here we look at the much humbler setting where . Recalling that each ordinal equals the set of its predecessors, and using interval notation to describe sets of ordinals, the Haddad-Sabbagh result is as follows:
Lemma. For all positive integers there exists a positive integer such that for any red-blue coloring of the edges of , and such that is blue, there is a subset of with monochromatic, and either is blue and , or else is red, , and .
Denote by the smallest number witnessing the lemma.
Theorem. If are positive integers, then , where .
The theorem was announced in 1969, but the proof never appeared. I have written a survey on the topic, including what should be the first proof in print of this result.
A topological variant of the function is obtained by requiring that the homogeneous set be a faithful topological copy of the corresponding ordinal. The resulting function we denote . Formally, iff is least such that any for red-blue coloring of there is an such that is monochromatic and, if is red, then is order-homeomorphic to , or if is blue, then is order-homeomorphic to .
The notion of order-homeomorphism is a natural one, the name itself is due to Baumgartner. We say that a set of ordinals is order-homeomorphic to an ordinal iff, first of all, has order type and, second, the unique order-isomorphism between and is a homeomorphism. Here, ordinals are equipped with the order topology, and carries the subspace topology (it is easy to check that if , then the topology inherited by as a subspace of is the same as the topology inherited as a subspace of . More concretely, the topological component of the definition simply assert that, for any limit ordinal , the th element of is the supremum of its predecessors in . That is, is closed as a subset of its supremum.
To illustrate the difference between and , let me mention that but , and but .
Theorem. The ordinal is countable for all finite .
The proof provides concrete upper bounds. The argument proceeds by induction, the key technical tool being the function that allows us to study the closed version of the pigeonhole principle:
One can prove that for all there is a such that whenever , either contains a subset order-homeomorphic to , or else contains a subset order-homeomorphic to . The smallest possible with this property we denote .
For example, , and . To verify that , what one actually proves is that .