arXiv subjects

Gian Marco Osso

Publications and source records attributed to Gian Marco Osso.

5 recordsLinked to original sources

arXiv2026

Computable aspects of the Laver partition theorem

The Laver Partition Theorem is a fundamental tool in the analysis of Laver and Hechler forcings. It is also connected to determinacy and the Galvin-Prikry theorem: indeed it can be seen as the common core of these two theorems. We study the reverse mathematics and Weihrauch degrees of the Laver Partition Theorem restricted to open and clopen sets. We obtain upper and lower bounds on the proof theoretic strength of this result, as well as a precise picture of the (arithmetical) Weihrauch degrees of the problems related to it.

math.LO↗

arXiv2026

The reverse mathematics of Brooks' theorem

This is an analysis of the status of Brooks' Theorem, a celebrated result in graph coloring, from the point of view of Reverse Mathematics. We prove that the restriction of Brooks' theorem to bounded graphs of degree greater than or equal to $3$ is provable in $\mathsf{RCA}_0$, while the statement for arbitrary graphs is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$. Brooks' Theorem for degree $2$, even when restricted to bounded graphs, is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$.

math.LO↗

arXiv2026

Listing the hyperarithmetical functions

Given a countable Turing ideal $\mathcal{I} \subseteq ω^ω$, we say that $x$ is a list (resp. weak list) of $\mathcal{I}$ if $\mathcal{I}=\{x^{[n]} : n \in ω\}$ (resp. if $\mathcal{I} \subseteq \{x^{[n]} :n \in ω\}$). We show that, for several natural ideals $\mathcal{I}$, $x$ computes a list of $\mathcal{I}$ if and only if it computes a function dominating all the functions in $\mathcal{I}$. On the other hand, we provide reals which are $\mathsf{HYP}$-strongly null engulfing (and hence $\mathsf{HYP}$-dominating, by results of Greenberg, Kuyper and Turetsky) but which cannot compute a weak list for $\mathsf{HYP}$, solving a problem left open in a recent paper by Greenberg and the second author. This result can be generalized to any countable ideal which is downward closed under $\leq_{\mathsf{HYP}}$. We also give a characterization of reals which compute a list of $\mathsf{HYP}$: $x$ computes a list of $\mathsf{HYP}$ if and only if $x$ is $\mathsf{HYP}$-dominating and $\mathcal{O}$ is $Σ^0_2(x)$.

math.LO↗

arXiv2026

Forcing and classes of $\mathsf{HYP}$-dominating functions

We study the question, what computational power is sufficient to perform constructions using either Laver or Hechler forcing. As a result, we obtain a separation between three relativised non-lowness classes that are the computability-theoretic analogues of three of the cardinals in Cichon's diagram.

math.LO↗

arXiv2024

The Galvin-Prikry Theorem in the Weihrauch lattice

This paper classifies different fragments of the Galvin-Prikry theorem, an infinite dimensional generalization of Ramsey's theorem, in terms of their uniform computational content (Weihrauch degree). It can be seen as a continuation of arXiv:2003.04245v3, which focused on the Weihrauch classification of functions related to the open (and clopen) Ramsey theorem. We show that functions related to the Galvin-Prikry theorem for Borel sets of rank n are strictly between the (n+1)-th and n-th iterate of the hyperjump operator $\mathsf{HJ}$, which is in turn equivalent to the better known $\widehat{\mathsf{WF}}$, which corresponds to $Π^1_1$-$\mathsf{CA}_0$ in the Weihrauch lattice. To establish this classification we obtain the following computability theoretic result: a Turing jump ideal containing homogeneous sets for all $Δ^0_{n+1}(X)$ sets must also contain the n-th hyperjump of X. We also extend our analysis to the transfinite levels of the Borel hierarchy. We further obtain some results about the reverse mathematics of the lightface fragments of the Galvin-Prikry theorem.

math.LO↗