Roma Logica: The Forum
Reverse Mathematics and Ramsey-type principles
Thursday Afternoon and Friday
October 29 and 30, 2026
Updated: September 29
The workshop will focus on Reverse Mathematics and in particular, the computability-theoretic and proof-theoretic strength of combinatorial Ramsey-type principles such as Ramsey’s Theorem for pairs, Hindman’s Theorem, and other similar statements.
The Colosseum was not available but we are a block away at ND Rome. We have one lecture room that sits about 40. There are 2 small white boards and the room is long. So board talks are not suggested.
Registration. Everyone needs to register. It is very quick. In particular we need a head count for the reception a week prior.
The organizers are Peter Cholak and Lorenzo Carlucci.
Schedule
Thursday, October 29
1:00 pm - 1:30 pm: Arrival and Registration
1:30 pm - 2:30 pm: Paul Shafer, Bounded Ramsey's theorem for triples in computability theory, Abstract below.
2:30 pm - 3:00 pm: Coffee Break
3:00 pm - 4:00 pm: Andrea Volpi, Ramsey principles for strong coloring classes and weak homogeneity conditions, Abstract below
4:15 pm - 5:15 pm: Mauro Di Nasso, A simultaneous generalization of Ramsey, Hindman, and
Hales-Jewett's Theorems, Abstract.
5:15 pm - 7:00pm: Heavy reception
Friday, October 30
9:30 am - 10:30 am: Gavin Dooley, Initial segment jump inversion for chains, Abstract Below.
10:30 am - 11:00 am: Coffee Break
11:00 am - 12:00 pm: Quentin Le-Houérou, The reverse mathematics of the bounded Ramsey theorem for pairs, Abstract Below.
12:00 pm - 2:00 pm: Lunch Break (on your own)
2:00 pm - 3:00 pm: Giordano Celli, A Weihrauch reduction from a combinatorial theorem to a well-ordering principle, Abstract Below.
3:00 pm - 3:30 pm: Coffee Break
3:30 pm - 4:30 pm: Alberto Marcone, The effective content of Laver partition theorem, Abstract below.
4:30 pm - ??, drinks at a nearby bar.
Travel Arrangements: Here you are on your own. Here are some nearby hotel and dining suggestions by the Notre Dame Global Gateway staff. Public transportation in Rome is good and you should be able to get a hotel or room elsewhere and travel to ND Rome from within Rome.
Many thanks to our two sponsors: Notre Dame Global and the Notre Dame Department of Mathematics.
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Giordano Celli, A Weihrauch reduction from a combinatorial theorem to a well-ordering principle.
In terms of Weihrauch complexity, the Ordered Ramsey's Theorem can be written as the parallel product of the Eventually Constant palette Tail principle and the well-ordering principle for the operator X^omega. We present a reduction from the first factor to the second.
Gavin Dooley, Initial segment jump inversion for chains.
We outline a forthcoming proof of a conjecture of Simpson asserting that, for any chain of Turing degrees 0' ≤ A₀ ≤ A₁ ≤ A₂ ≤ …, there exists an initial segment of Turing degrees 0 < M₀ < M₁ < M₂ < … such that Mᵢ' ≡ Aᵢ for each i. As a corollary, we deduce that COH does not imply FIP over ω-models, answering a question of Patey and clarifying the relationship between Ramsey's theorem for pairs and the axiom of choice.
Quentin Le-Houérou, The reverse mathematics of the bounded Ramsey theorem for pairs.
The bounded Ramsey theorem for pairs (BRT^2_2) is a weakening of Ramsey's theorem for pairs (RT^2_2), where we have a guarantee that no clique of color 0 for a given size k exists. This weakening is computably true, in the sense that every computable instance of it has a computable solution, and we show that proving this fact requires Σ_2-induction. On the other hand, the mere existence of a solution (not necessarily a computable one) is a direct consequence of RT^2_2, which is known not imply IΣ_2 over RCA_0. This is a joint work with Ludovic Patey.
Alberto Marcone, The effective content of Laver partition theorem.
This talk deals with the effective content of a partition theorem for Laver trees, in the context of reverse mathematics and Weihrauch reducibility. Laver forcing, and its variants, are an important tool to prove e.g. consistency results about cardinal characteristics of the continuum. A key feature of Laver forcing is "pure decision", which can be stated as a Ramsey-like statement which follows, level by level, from both determinacy and the Ramsey property. We investigate the clopen and open cases. Joint work with Gian Marco Osso.
Paul Shafer, Bounded Ramsey's theorem for triples in computability theory
Bounded Ramsey's theorem for 2-colorings of n-tuples restricts RT^n_2 to colorings where the homogeneous sets for color 1 are of bounded size. We investigate the computational content of BRT^3_(2,l), which states that every 2-coloring f of N with no homogeneous set for color 1 of size l > 3 has an infinite homogeneous set for color 0. Frittaion initiated the systematic study of bounded Ramsey's theorem. Soldà picked up the topic, proving in his thesis that the statement "BRT^3_(2,l) holds for every l" is cone avoiding and hence does not imply ACA_0. Recently, Le Houérou and Patey studied BRT^2_(2,l) for l > 2.
It is not difficult to see that RT^2_2 computably reduces to BRT^3_(2,4). Thus bounded Ramsey's theorem for triples is at least as complicated as Ramsey's theorem for pairs. We find that bounded Ramsey's theorem for triples is computationally similar to Ramsey's theorem for pairs despite converse reductions failing. For example, the statement "BRT^3_(2,l) holds for every l" admits constant-bounded trace avoidance and a weakly low basis, just like the statement "RT^2_l holds for every l." However, BRT^3_(2,4) does not reduce to RT^2_2 via a k-move reduction game (à la Hirschfeldt and Jockusch) for any k. Furthermore, BRT^3_(2,4) does not computably reduce to the statement "RT^2_l holds for every l."
Andrea Volpi, Ramsey principles for strong coloring classes and weak homogeneity conditions
There is a proof that Ramsey's theorem for triples and $2$ colors implies $\ACA$ that uses a weaker notion of homogeneity called path homogeneity. A set $H = \{h_0 < h_1 < \ldots < \}$ is path homogeneous for a coloring $c$ if for all $i$, $(h_i, h_{i+1}, h_{i+2})$ has the same $c$-color. Starting from this remark, we study different Ramsey-like statements depending on restrictions on the colorings considered and the notion of homogeneity required. Some of this notions already appear in the framework of reverse mathematics under different names. For instance the ascending descending principle $\ADS$ is exactly Ramsey's theorem for pairs and 2 colors restricted to transitive colorings where solutions are homogeneous sets, while Erdős–Moser theorem $\EM$ is Ramsey's theorem for pairs and 2 colors where the solution is an infinite set where the coloring is transitive. We define and study generalizations of these and other principles for $n$-tuples where $n \ge 3$ and more than $2$ colors in the framework of reverse mathematics. This is joint work with Lorenzo Carlucci and Oriola Gjetaj.