Related papers: Ramsey Theory for Words Representing Rationals
We give a parametrization with perfect subsets of $2^{\infty}$ of the abstract Ramsey theorem (see \cite{todo}) Our main tool is an extension of the parametrized version of the combinatorial forcing developed in \cite{nash} and \cite{todo},…
We give a characterizations of Ramsey ultrafilters on $\mathscr P(\omega)$ in terms of functions $f:\omega^n\to\omega$ and their ultrafilter extensions. To do this, we prove that for any partition $\mathcal P$ of $[\omega]^n$ there is a…
These lecture notes provide an introduction to combinatorics on words and its interactions with dynamics, algebra, and arithmetic. The central theme is the notion of low factor complexity for infinite words. We investigate the following…
We prove that double exponentiation is an upper bound to Ramsey theorem for colouring of pairs when we want to predetermine the order of the differences of successive members of the homogeneous set.
We prove some results on the border of Ramsey theory (finite partition calculus) and model theory. Also a beginning of classification theory of finite models in undertaken.
We present a method for the enumeration of restricted words over a finite alphabet. Restrictions are described through the inclusion or exclusion of suitable building blocks used to construct the words by concatenation. Our approach, which…
We prove a density version of the Carlson--Simpson Theorem. Specifically we show the following. For every integer $k\geq 2$ and every set $A$ of words over $k$ satisfying \[\limsup_{n\to\infty} \frac{|A\cap [k]^n|}{k^n}>0\] there exist a…
Given the set of words of a given length for a given alphabet, the Hamming metric between two such words is the number of positions where the two words differ. A quantum version of the corresponding Kantorovich-Wasserstein metric on states…
It is observed that the conjugacy growth series of the infinite finitary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed,…
We consider the set of finite random words $\mathcal A^\star$, with independent letters drawn from a finite or infinite totally ordered alphabet according to a general probability distribution. On a specific subset of $\mathcal A^\star$,…
We identify computability-theoretic properties enabling us to separate various statements about partial orders in reverse mathematics. We obtain simpler proofs of existing separations, and deduce new compound ones. This work is part of a…
The theory of finite term algebras provides a natural framework to describe the semantics of functional languages. The ability to efficiently reason about term algebras is essential to automate program analysis and verification for…
We introduce the notion of a rational dynamical system extending the classical notion of a topological dynamical system and we prove (multiple) recurrence results for such systems via a partition theorem for the rational numbers proved by…
We study the strength of axioms needed to prove various results related to automata on infinite words and B\"uchi's theorem on the decidability of the MSO theory of $(N, {\le})$. We prove that the following are equivalent over the weak…
We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a…
We contribute to the refined understanding of the language-logic-algebra interplay in the context of first-order properties of countable words. We establish decidable algebraic characterizations of one variable fragment of FO as well as…
We introduce and study some families of groups whose irreducible characters take values on quadratic extensions of the rationals. We focus mostly on a generalization of inverse semi-rational groups, which we call uniformly semi-rational…
We resolve a conjecture of Cooper-Fenner-Purewal that a certain sequence of combinatorial matrices which can be used to bound small product-Ramsey numbers is positive semidefinite. Because the connection to Ramsey Theory involves solving…
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…
We show that the Ramsey theory of block sequences in infinite-dimensional discrete vector spaces can be parametrized by perfect sets. As special cases, we prove combinatorial dichotomies for definable families of partitions and linear…