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Mauro Di Nasso

Publications and source records attributed to Mauro Di Nasso.

At least 19 recordsLinked to original sources
arXiv2026

Hypernatural numbers in arithmetic Ramsey theory

The hypernatural numbers ${}^*\mathbb{N}$ of nonstandard analysis have recently proven to be an effective tool in arithmetic Ramsey theory. After introducing the fundamental ``nonstandard" notions, we present several examples to illustrate the use of this technique in practice. In particular, we provide brief nonstandard proofs of some recent results concerning the monochromaticity of certain families of finite and infinite configurations.

math.NT↗
arXiv2026

A Van der Waerden-free proof of Rado's theorem

We present a proof of the sufficiency of Rado's condition for the partition regularity of linear Diophantine equations that avoids any use of van der Waerden's theorem. The proof is based on fundamental properties that are common knowledge in combinatorics of numbers and is entirely elementary, with the sole exception of a standard application of the compactness principle.

math.CO↗
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

A simultaneous extension of Ramsey, Hindman, and Hales-Jewett Theorems

We prove a multidimensional extension of a strong Hales-Jewett theorem that simultaneously and "directly" extends Ramsey's theorem and Hindman's theorem. The proofs show the effectiveness and simplicity of the techniques based on iterated nonstandard extensions that have been recently developed. Unlike existing ultrafilter proofs, our arguments to prove the strong Hales-Jewett theorem assume neither minimal nor idempotent ultrafilters. To demonstrate this, we translate our proof of the strong Hales-Jewett theorem into an ultrafilter proof that requires only non-principal ultrafilters.

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

A new ultrafilter proof of Van der Waerden's theorem

We present a new short proof of Van der Waerden's Theorem about the existence of arbitrarily long monochromatic arithmetic progressions. The proof uses algebra in the compact space of ultrafilters $β\N$, but contrarily to the other existing proofs, neither minimal nor idempotent ultrafilters are involved.

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

The magic of tensor products of ultrafilters

Tensor products of ultrafilters have special combinatorial features closely related to Ramsey's Theorem, making them useful tools in applications. Here we first review their fundamental properties and isolate some new ones, including a characterisation of the limit superior and inferior of sequences as limits along tensor products, and an application to the Banach asymptotic density. We then prove a general result on the combinatorial structure of sets belonging to tensor products and, as a result, we obtain several characterisations of the additive properties of sets of natural numbers. Finally, we show that tensor products can be described as idempotents of an appropriate semigroup.

math.CO↗
arXiv2024

Foundations of iterated star maps and their use in combinatorics

We develop a framework for nonstandard analysis that gives foundations to the interplay between external and internal iterations of the star map, and we present a few examples to show the strength and flexibility of such a nonstandard technique for applications in combinatorial number theory.

math.CO↗
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↗
arXiv2022

Euclidean integers, Euclidean ultrafilters, and Euclidean numerosities

We introduce axiomatically the ring $\bf{Z}_κ$ of the Euclidean integers, that can be viewed as the ``integral part" of the field $\mathbb{E}$ of Euclidean numbers of [4], where the transfinite sum of ordinal indexed $κ$-sequences of integers is well defined. In particular any ordinal might be identified with the transfiite sum of its characteristic function, preserving the so called natural operations. The ordered ring $\bf{Z}_κ$ may be obtained as an ultrapower of $\mathbb{Z}$ modulo suitable ultrafilters, thus constituting a \it{ring of nonstandard integers.} Most relevant is the \it{algebraic} characterization of the ordering: a Euclidean integer is \it{positive} if and only if it is \it{the transfinite sum of natural numbers.} This property requires the use of special ultrafilters called Euclidean, here introduced to ths end. The ring $\bf{Z}_κ$ allows to assign a ``Euclidean" size (\it{numerosity}) to ``ordinal Punktmengen", i.e. sets of tuples of ordinals, as the transfinite sum of their characteristic functions: so every set becomes equinumerous to a set of ordinals, the Cantorian defiitions of \it{order, addition and multiplication} are maintained, while the Euclidean principle ``the whole is greater than the part" (\it{a set is (strictly) larger than its proper subsets}) is fulfilled.

math.LO↗
arXiv2022

Infinite monochromatic patterns in the integers

We show the existence of several infinite monochromatic patterns in the integers obtained as values of suitable symmetric polynomials. The simplest example is the following. For every finite coloring of the natural numbers $\mathbb{N}=C_1\cup\ldots\cup C_r$, there exists an increasing sequence $a<b<c<\ldots$ such that all elements below are monochromatic, that is, they belong to the same $C_i$: $$a,b,c,\ldots, a+b+ab, a+c+ac, b+c+bc,\ldots,a+b+c+ab+ac+bc+abc,\ldots.$$ The proofs use algebra in the space of ultrafilters $β\mathbb{Z}$.

math.CO↗
arXiv2020

Translation invariant filters and van der Waerden's Theorem

We present a self-contained proof of a strong version of van der Waerden's Theorem. By using translation invariant filters that are maximal with respect to inclusion, a simple inductive argument shows the existence of "piecewise syndetically"-many monochromatic arithmetic progressions of any length k in every finite coloring of the natural numbers. All the presented constructions are constructive in nature, in the sense that the involved maximal filters are defined by recurrence on suitable countable algebras of sets. No use of the axiom of choice or of Zorn's Lemma is needed.

math.CO↗
arXiv2017

Abstract densities and ideals of sets

Abstract upper densities are monotone and subadditive functions from the power set of positive integers to the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper Banach density, and the upper logarithmic density. We answer a question posed by G. Grekos in 2013, and prove the existence of translation invariant abstract upper densities onto the unit interval, whose null sets are precisely the family of finite sets, or the family of sequences whose series of reciprocals converge. We also show that no such density can be atomless. (More generally, these results also hold for a large class of summable ideals.)

math.NT↗
arXiv2017

Idempotent ultrafilters without Zorn's Lemma

We introduce the notion of additive filter and present a new proof of the existence of idempotent ultrafilters on N without any use of Zorn's Lemma, and where one only assumes the Ultrafilter Theorem for the continuum.

math.LO↗