Alberto Marcone
Order dimension theory measures the complexity of partially ordered sets by quantifying how far they are from being linearly ordered. In this paper we study classical bounding results for order dimension within the framework of reverse…
We study the existence and the distribution of "long" chains in the Weihrauch degrees, mostly focusing on chains with uncountable cofinality. We characterize when such chains have an upper bound and prove that there are no cofinal chains…
We prove the computability of a version of Whitney Extension, when the input is suitably represented. More specifically, if $F \subseteq \mathbb{R}^n$ is a closed set represented so that the distance function $x \mapsto d(x,F)$ can be…
We solve a long-standing open problem, formulated by Krasner in the 1950's, in the context of Polish (i.e. separable complete) ultrametric spaces by providing a characterization of their isometry groups using suitable forms of generalized…
In this paper we study a very general finite Ramsey theorem, where both the sets being colored and the homogeneous set must satisfy some largeness notion. For the homogeneous set this has already been done using the notion of…
We introduce and study a multiplicative analogue of additive indecomposability for linear order types that we call untranscendability, as well as a strengthening that we call $s$-untranscendability. We show that, with the unique exception…
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…
A partial order $(P,\le)$ admits a jump operator if there is a map $j\colon P \to P$ that is strictly increasing and weakly monotone. Despite its name, the jump in the Weihrauch lattice fails to satisfy both of these properties: it is not…
This paper continues the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. In particular, we consider problems related to perfect subsets of Polish spaces, studying the perfect set…
Given a nonempty set $\mathcal{L}$ of linear orders, we say that the linear order $L$ is $\mathcal{L}$-convex embeddable into the linear order $L'$ if it is possible to partition $L$ into convex sets indexed by some element of $\mathcal{L}$…
We consider countable linear orders and study the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results, and further extend our research to the case of…
Recently, a surprising connection between algorithmic learning of algebraic structures and descriptive set theory has emerged. Following this line of research, we define the learning power of an equivalence relation $E$ on a topological…
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…
We analyze the axiomatic strength of the following theorem due to Rival and Sands in the style of reverse mathematics. "Every infinite partial order $P$ of finite width contains an infinite chain $C$ such that every element of $P$ is either…
We investigate the uniform computational content of the open and clopen Ramsey theorems in the Weihrauch lattice. While they are known to be equivalent to $\mathrm{ATR_0}$ from the point of view of reverse mathematics, there is not a…
It has recently been shown that fairly strong axiom systems such as $\mathsf{ACA}_0$ cannot prove that the antichain with three elements is a better quasi order ($\mathsf{bqo}$). In the present paper, we give a complete characterization of…
In this paper, we study Hausdorff and Fourier dimension from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity. Working in the hyperspace $\mathbf{K}(X)$ of compact subsets of $X$, with $X=[0,1]^d$ or…
Interval graphs and interval orders are deeply linked. In fact, edges of an interval graphs represent the incomparability relation of an interval order, and in general, of different interval orders. The question about the conditions under…
In this paper we study the notion of Salem set from the point of view of descriptive set theory. We first work in the hyperspace $\mathbf{K}([0,1])$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a…
We define an on-line (incremental) algorithm that, given a (possibly infinite) pseudo-transitive oriented graph, produces a transitive reorientation. This implies that a theorem of Ghouila-Houri is provable in RCA_0 and hence is computably…