Foundations of Mathematics

February 1, 2019

Foundations of Mathematics, Andrés E. Caicedo, James Cummings, Peter Koellner, and Paul B. Larson, eds., Contemporary Mathematics, vol. 690, Amer. Math. Soc., Providence, RI, 2017. DOI: 10.1090/conm/690. MR3656304. Zbl 06733965.

This book contains the proceedings of the conference in honor of Hugh Woodin’s 60th birthday, that I previously discussed on this blog (here, here, and here).

The AMS page for the volume can be found here, including the table of contents and links to the front- and endmatter (which I think are available to everybody) and links to the individual papers (which I imagine may not be).

Topological Ramsey numbers and countable ordinals

February 1, 2019

Paul Erdős and Eric Milner published in 1972 A theorem in the partition calculus, where they established that if $\beta$ is a countable ordinal and $n\in\omega$, then there is a countable ordinal $\alpha$ such that $\alpha\to(\beta,n)^2,$

meaning that any graph whose set of vertices is $\alpha$ either contains a clique (complete subgraph) whose set of vertices $H$ has order type $\beta$ or an independent set of size $n$.

The result is false if $n$ is replaced by $\omega$, except for when $\beta=\omega$, in which case we can take $\alpha=\omega$ as well, this is Ramsey’s theorem.

The least $\alpha$ such that $\alpha\to(\omega+1,\omega)^2$ is $\alpha=\omega_1$, in which case a stronger result holds, namely $\omega_1\to(\omega_1,\omega+1)^2$. In fact, more is true: the homogeneous set $H$ of order type $\omega_1$ can be taken to be a stationary subset of $\omega_1$, and the set of type $\omega+1$ can be required to be closed, meaning that its $\omega$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 $\omega_1\to_{cl}(\beta,\omega+1)^2$ holds for any countable ordinal $\beta$, where the subindex cl indicates that the sets of vertices of type $\beta$ or $\omega+1$ 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 $\alpha$ such that $\alpha\to_{cl}(\beta,n)^2$ for specific values of $\beta<\omega_1$ and $n<\omega$. 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.

Partiti

February 1, 2019

Partiti is a puzzle created by Thinh Van Duc Lai, a Vietnamese puzzle enthusiast most of whose puzzles involve mathematics in some form. His work has been featured in The New York Times, see here. Partiti puzzles appeared on Mathematics Magazine throughout 2018.

Brittany Shelton and I coauthored a short piece introducing the puzzle to the Magazine readers at the invitation of Michael Jones, the magazine editor and a colleague at Mathematical Reviews. It is titled Of puzzles and partitions. Introducing Partiti, and can be found on the arXiv or through my papers page.

The fourteen Victoria Delfino problems and their status in the year 2019

February 1, 2019

The Cabal seminar in southern California was instrumental to the development of determinacy. The Delfino problems were suggested as a way to measure progress on this area. Fourteen problems were suggested in total through the years. Some were solved very quickly after their proposal, a few remain open.

Benedikt Löwe and I wrote a survey of their current status, cleverly titled The fourteen Victoria Delfino problems and their status in the year 2019. The cleverness has forced us to keep changing its title as its publication date kept being postponed. It is scheduled to appear in the fourth volume of the reissued Cabal volumes, which I am told is expected to finally be published this year. The volumes are being published by the Association for Symbolic Logic and Cambridge University Press as part of the Lecture Notes in Logic series.

The survey can be accessed through the Hamburger Beiträge zur Mathematik preprint server; it is paper 770 there. It can also be found through my papers page (currently under notes, and later on, once it appears, under papers).