arXiv subjects

Mariaclara Ragosta

Publications and source records attributed to Mariaclara Ragosta.

6 recordsLinked to original sources

arXiv2026

Monochromatic exponential triples: an ultrafilter proof

We present a short ultrafilter proof of the existence of monochromatic exponential triples $\{a, b, b^a\}$ in any finite coloring of the natural numbers. The proof is given from scratch and uses only Ramsey's theorem, the notion of asymptotic density and the definition of ultrafilter as prerequisites. We then generalize the construction using a special ultrafilter whose existence is well known in the algebra of ultrafilters, and prove a new result on the existence of infinite monochromatic exponential patterns.

math.CO↗

arXiv2026

Central sets and infinite monochromatic exponential patterns

A cornerstone of Arithmetic Ramsey Theory is \emph{Hindman's Theorem} of 1974: "For every finite coloring of the natural numbers, there exists an infinite sequence $(x_n)$ such that all finite sums $x_{n_1}+\ldots+x_{n_k}$ of distinct elements are monochromatic". We extend the validity of Hindman's theorem to a broad class of non-associative operations that generalize exponentiation between natural numbers. The main tool we use in our proofs is given by the central sets, a special class of sets isolated in 1981 by H. Furstenberg in the context of topological dynamics. It was later discovered in 1990 by V. Bergelson and N. Hindman that central sets can be characterized as those sets that belong to a minimal idempotent ultrafilter, thus opening up the study of their rich combinatorial structure with the well-developed machinery of algebra in the space of ultrafilters. As a corollary of our main result, we obtain an extension of Sahasrabudhe's results of 2018 about the existence of arbitrarily large (but finite) monochromatic exponential patterns; indeed, we obtain the existence of an infinite sequence such that all finite exponential configurations originating from its elements are monochromatic.

math.CO↗

arXiv2026

Monochromatic sums and quotients in $\mathbb N$

We prove partition regularity of the configuration $x,y,x+y,y/x$ in a strong infinitary form that extends Hindman's Theorem. We study the related issue of partition regularity of configurations involving products of a degree one polynomial in $x$ with one in $y$, reducing the general problem to a handful of special cases.

math.LO↗

arXiv2026

Ramsey's witnesses

We introduce the notion of Ramsey partition regularity, a generalisation of partition regularity involving infinitary configurations. We provide characterisations of this notion in terms of certain ultrafilters related to tensor products and dubbed Ramsey's witnesses; and we also consider their nonstandard counterparts as pairs of hypernatural numbers, called Ramsey pairs. These characterisations are then used to determine whether various configurations involving polynomials and exponentials are Ramsey partition regular over the natural numbers. In particular, this provides negative answers to several questions recently posed by Kra, Moreira, Richter and Robertson.

math.LO↗

arXiv2025

Extending orders to types

Given an ordered structure, we study a natural way to extend the order to preorders on type spaces. For definably complete, linearly ordered structures, we give a characterisation of the preorder on the space of 1-types. We apply these results to the divisibility preorder on the space of ultrafilters on the set of natural numbers, giving an independence result about the suborder consisting of ultrafilters with only one fixed prime divisor, as well as a classification of ultrafilters with finitely many prime divisors.

math.LO↗

arXiv2023

Self-divisible ultrafilters and congruences in $β\mathbb{Z}$

We introduce self-divisible ultrafilters, which we prove to be precisely those $w$ such that the weak congruence relation $\equiv_w$ introduced by Šobot is an equivalence relation on $β\mathbb{Z}$. We provide several examples and additional characterisations; notably we show that $w$ is self-divisible if and only if $\equiv_w$ coincides with the strong congruence relation $\equiv^{\mathrm{s}}_{w}$, if and only if the quotient $(β\mathbb{Z},\oplus)/\mathord{\equiv^{\mathrm{s}}_w}$ is a profinite group. We also construct an ultrafilter $w$ such that $\equiv_w$ fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion $\hat{\mathbb{Z}}$ of the integers.

math.LO↗