arXiv subjects

Luke Serafin

Publications and source records attributed to Luke Serafin.

8 recordsLinked to original sources

arXiv2026

Projective Chromatic Numbers

We extend classical notions of definable colourability of graphs to the general projective setting and investigate whether known results, mainly about the $G_0$ dichotomy and the $2n + 1$ conjecture, hold in the context of higher projective pointclasses. We establish that for $n \ge 2$, the presence of a $\mathbf{\Delta}^1_n$-definable well-order of the reals implies $\chi_{\mathbf{\Delta^1_n}}(G) = \chi(G)$ for all locally countable $\mathbf{\Delta^1_n}$-definable graphs $G$, and that the presence of a $\mathbf{\Delta^1_2}$-definable well-order of the reals implies $\chi_{\mathbf{\Delta^1_2}}(G) = \chi(G)$ for all locally countable Borel graphs $G$.

math.LO↗

arXiv2026

Funicular preorders can be prelinearized without nonprincipal ultrafilters over $\mathbb N$

It is a consequence of the axiom of choice that every preorder can be extended to a total preorder while respecting the strict preorder relation. We call such an extension a prelinearization of the preorder and study the extent to which the axiom of choice is needed to construct prelinearizations. We isolate the class of funicular preorders, and show that these have prelinearizations in models of $\mathsf{ZF+DC}$ containing no nonprincipal ultrafilters over $\omega$. Funicular preorders include coordinatewise domination on $\mathbb{R}^\omega$, Turing reducibility, and various preorders arising in social choice theory. The relevant models are constructed first using tools from the geometric set theory of Larson and Zapletal, which requires an inaccessible cardinal, and then the inaccessible cardinal is eliminated using methods of amalgamation for Cohen reals.

math.LO↗

arXiv2025

Morita Rigidity for Kleene Algebras

We introduce Morita equivalence to the study of Kleene algebras and modules. Classical characterizations of Morita-equivalent semirings such as having equivalent categories of modules and one semiring being a full matrix algebra over the other carry over. We also observe that Morita equivalence can be applied to extending and restricting scalars in Lindenbaum Tarski algebras of propositional dynamic logics. But the signature result which we obtain is a form of rigidity for Kleene algebras, which states that if the semiring reducts of two Kleene algebras are Morita-equivalent, then the Morita equivalence is in fact witnessed by Kleene bimodules.

cs.LO↗

arXiv2025

Ultrafilters over Successor Cardinals and the Tukey Order

We study ultrafilters on regular uncountable cardinals, with a primary focus on $\omega_1$, and particularly in relation to the Tukey order on directed sets. Results include the independence from ZFC of the assertion that every uniform ultrafilter over $\omega_1$ is Tukey-equivalent to $[2^{\aleph_1}]^{<\omega}$, and for each cardinal $\kappa$ of uncountable cofinality, a new construction of a uniform ultrafilter over $\kappa$ which extends the club filter and is Tukey-equivalent to $[2^\kappa]^{<\omega}$. We also analyze Todorcevic's ultrafilter $\mathcal{U}(T)$ under PFA, proving that it is Tukey-equivalent to $[2^{\aleph_1}]^{<\omega}$ and that it is minimal in the Rudin-Keisler order with respect to being a uniform ultrafilter over $\omega_1$. We prove that, unlike PFA, $\text{MA}_{\omega_1}$ is consistent with the existence of a coherent Aronszajn tree $T$ for which $\mathcal{U}(T)$ extends the club filter. A number of other results are obtained concerning the Tukey order on uniform ultrafilters and on uncountable directed systems.

math.LO↗

arXiv2024

On Strongly-equitable Social Welfare Orders Without the Axiom of Choice

Social welfare orders seek to combine the disparate preferences of an infinite sequence of generations into a single, societal preference order in some reasonably-equitable way. In [2] Dubey and Laguzzi study a type of social welfare order which they call SEA, for strongly equitable and (finitely) anonymous. They prove that the existence of a SEA order implies the existence of a set of reals which does not have the Baire property, and observe that a nonprincipal ultrafilter over $\mathbb{N}$ can be used to construct a SEA order. Questions arising in their work include whether the existence of a SEA order implies the existence of either a set of real numbers which is not Lebesgue-measurable or of a nonprincipal ultrafilter over $\mathbb{N}$. We answer both these questions, the solution to the second using the techniques of geometric set theory as set out by Larson and Zapletal in [11]. The outcome is that the existence of a SEA order does imply the existence of a set of reals which is not Lebesgue-measurable, and does not imply the existence of a nonprincipal ultrafilter on $\mathbb{N}$.

math.LO↗

arXiv2023

Ultrafilters, Transversals, and the Hat Game

Geschke, Lubarsky, and Rahn in ``Choice and the Hat Game''~\cite{choice-and-the-hat-game} generalize the classic hat game puzzle to infinitely-many players and ask whether every model of set theory without choice in which the optimal solution can be carried out contains either a nonprincipal ultrafilter on $\mathbb N$ or else a Vitali set. A negative answer is obtained here by constructing a model in which there is an optimal solution to the hat game puzzle but no nonprincipal ultrafilter on $\mathbb N$ and no Vitali set. This is accomplished in a more general setting, establishing that for any Borel bipartite graph $Γ$ not embedding some $K_{n,ω_1}$ and with countable colouring number there is a model of $\mbox{ZF} + \mbox{DC}$ in which $Γ$ has a $2$-colouring but there is no ultrafilter as above or Vitali set. The same conclusion applies to the natural generalization of the hat game to an arbitrary finite number of hat colours.

math.LO↗

arXiv2017

A formally verified proof of the Central Limit Theorem

We describe a proof of the Central Limit Theorem that has been formally verified in the Isabelle proof assistant. Our formalization builds upon and extends Isabelle's libraries for analysis and measure-theoretic probability. The proof of the theorem uses characteristic functions, which are a kind of Fourier transform, to demonstrate that, under suitable hypotheses, sums of random variables converge weakly to the standard normal distribution. We also discuss the libraries and infrastructure that supported the formalization, and reflect on some of the lessons we have learned from the effort.

cs.MS↗

arXiv2015

On generalizations of separating and splitting families

The work in this article is concerned with two different types of families of finite sets: separating families and splitting families (they are also called "systems"). These families have applications in combinatorial search, coding theory, cryptography, and related fields. We define and study generalizations of these two notions, which we have named $n$-separating families and $n$-splitting families. For each of these new notions, we outline their basic properties and connections with the well-studied notions. We then spend the greatest effort obtaining lower and upper bounds on the minimal size of the families. For $n$-separating families we obtain bounds which are asymptotically tight within a linear factor. For $n$-splitting families this appears to be much harder; we provide partial results and open questions.

math.CO↗