arXiv subjects

Tonatiuh Matos-Wiederhold

Publications and source records attributed to Tonatiuh Matos-Wiederhold.

6 recordsLinked to original sources

arXiv2026

Complexity of deep computations via topology of function spaces

We use topological methods to study complexity of deep computations and limit computations. We use topology of function spaces, specifically, the classification of Rosenthal compacta, to identify new complexity classes. We use the language of model theory, specifically, the concept of \emph{independence} from Shelah's classification theory, to translate between topology and computation. We use the theory of Rosenthal compacta to characterize approximablility of deep computations, both deterministically and probabilistically.

math.LO↗

arXiv2026

No Countable Basis for Borel Directed Graphs of Dichromatic Number at Least Three

I prove that the Borel directed graphs whose vertex set admits a partition into two Borel acyclic sets form a $\mathbfΣ^1_2$-complete set; equivalently, that deciding whether a Borel directed graph has Borel dichromatic number at least~$3$ is a $\mathbfΠ^1_2$-complete problem. It follows that no countable family of Borel directed graphs can serve as a basis for this class under Borel homomorphism and, more generally, that any basis must be at least as complex as~$\mathbfΠ^1_2$. The proof lifts the classical NP-completeness reduction of Bokal, Fijavž, Juvan, Kayll, and Mohar to the Borel setting, using the coding framework of Thornton. Combined with a straightforward reduction from undirected to directed coloring problems, this completes the picture for finite Borel chromatic and dichromatic thresholds: for every finite $k$, the set of Borel (directed) graphs admitting a Borel $k$-(di)coloring is $\mathbfΣ^1_2$-complete, and in particular admits no countable basis. This contrasts with the uncountable threshold, where a single-element basis exists for Borel chromatic number (Kechris--Solecki--Todorčević) and a continuum-size basis exists for Borel dichromatic number (Raghavan--Xiao).

math.LO↗

arXiv2026

A Concise Proof of the $L_0$ Dichotomy

Carroy, Miller, Schrittesser, and Vidnyánszky established the $L_0$ dichotomy: there is a Borel graph of Borel chromatic number three that admits a continuous homomorphism to every analytic graph of Borel chromatic number at least three. Their proof relies on a transfinite analysis of terminal approximations over a decreasing $ω_1$-sequence of analytic sets. I give a new, substantially shorter proof of this result by adapting the graph-theoretic framework recently introduced by Bernshteyn for the $G_0$ dichotomy. The central device is a $σ$-ideal of \emph{small} sets of homomorphisms from finite path approximations into the target graph, where smallness is witnessed by a bounded odd-walk condition on vertex projections. The key lemma that largeness is preserved under the doubling operation is established via the First Reflection Theorem, replacing the original transfinite construction with a single Borel reflection argument. The continuous homomorphism from the canonical graph $\mathbb{L}_c$ into the target is then obtained as a limit of shrinking families of copies, in direct analogy with Bernshteyn's proof for $G_0$.

math.LO↗

arXiv2025

The Open Coloring Axiom

This work is concerned with an axiom introduced by Todorcěvić in \cite{stevo} that constitutes a Ramsey-like statement regarding the topology of the reals. Our aim is to explain the axiom in detail, give some interesting applications and finally prove that the axiom is indeed consistent with ZFC, so that it makes sense to consider working with it in the first place. For this particular academic endeavor, we cover several advanced topics in set theory, including concepts like {\sl Hausdorff gaps}, forcing, infinitary combinatorics and a tad of topology. We employ, for example, an argument based on Rothberger's theorem to show that the Open Coloring Axiom implies the equality $\mathfrak b=\aleph_2$, which in turn makes this axiom inconsistent with CH. In other words, in ZFC, the Open Coloring Axiom could be false. To prove its relative consistency, we show that the axiom could be true by following a rather long and technical lemma of Todorcěvić, which leads to the culmination of this work.

math.LO↗

arXiv2024

Uncountable sets and an infinite linear order game

An infinite game on the set of real numbers appeared in Matthew Baker's work [Math. Mag. 80 (2007), no. 5, pp. 377--380] in which he asks whether it can help characterize countable subsets of the reals. This question is in a similar spirit to how the Banach-Mazur Game characterizes meager sets in an arbitrary topological space. In a recent paper, Will Brian and Steven Clontz prove that in Baker's game, Player II has a winning strategy if and only if the payoff set is countable. They also asked if it is possible, in general linear orders, for Player II to have a winning strategy on some uncountable set. To this we give a positive answer and moreover construct, for every infinite cardinal $κ$, a dense linear order of size $κ$ on which Player II has a winning strategy on all payoff sets. We finish with some future research questions, further underlining the difficulty in generalizing the characterization of Brian and Clontz to linear orders.

math.LO↗

arXiv2020

An alternative to back-and-forth

We exhibit how the Rasiowa-Sikorski Lemma simplifies, in a sense, proofs of results that make use of the technique known as back-and-forth, often resulting in not very illustrative arguments. The first two sections seek to show one simple and one complicated proofs of known results, in the hopes that the reader appreciates how the arguments end up, in our view, considerably clearer than those found in classic literature. The final section shows how the same techniques can be adapted to areas commonly considered distant to Set Theory, in this instance, Graph Theory. Sections one and two are based on my bachelor thesis, under the direction of Dr. Roberto Pichardo Mendoza, whom I deeply thank for his advice and revision of this work. All results mentioned in this paper are well known, however, as far as we know, the proofs in the last two sections are original.

math.LO↗