Related papers: Van der Waerden's Theorem and Avoidability in Word…
Free words are elements of a free monoid, generated over an alphabet via the binary operation of concatenation. Casually speaking, a free word is a finite string of letters. Henceforth, we simply refer to them as words. Motivated by recent…
A non-empty word $w$ is a border of the word $u$ if $\vert w\vert<\vert u\vert$ and $w$ is both a prefix and a suffix of $u$. A word $u$ with the border $w$ is closed if $u$ has exactly two occurrences of $w$. A word $u$ is privileged if…
We study the impact of certain identities and probabilistic identities on the structure of finite groups. More specifically, let $w$ be a nontrivial word in $d$ distinct variables and let $G$ be a finite group for which the word map…
Using a method we have utilized previously, namely through a finite power series expansion which also sometimes is known as the "radix polynomial" representation of an integer, we find an upper bound for a van der Waerden number that has a…
In the literature two notions of the word problem for a variety occur. A variety has a decidable word problem if every finitely presented algebra in the variety has a decidable word problem. It has a uniformly decidable word problem if…
Let A be an alphabet and W be a set of words in the free monoid A*. Let S(W) denote the Rees quotient over the ideal of A* consisting of all words that are not subwords of words in W. We call a set of words W finitely based if the monoid…
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…
A double occurrence word $w$ over a finite alphabet $\Sigma$ is a word in which each alphabet letter appears exactly twice. Such words arise naturally in the study of topology, graph theory, and combinatorics. Recently, double occurrence…
We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if $\{1,\dots,N\}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$…
Given a word $w$, what is the maximum possible number of appearances of $w$ reading contiguously along any of the directions in $\{-1, 0, 1\}^d \setminus \{\mathbf{0}\}$ in a large $d$-dimensional grid (as in a word search)? Patchell and…
The complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. We study infinite binary words $\bf w$ that avoid sufficiently large complementary factors; that is, if $x$ is a factor of…
A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern $p$ is unavoidable if, over every finite alphabet, every sufficiently long word encounters $p$. A theorem by…
We give a purely combinatorial proof for a two-fold generalization of van der Waerden-Brauer's theorem and Hindman's theorem. We also give tower bounds for a finite version of it.
Let $w$ be a word in the free group of rank $n \in \mathbb{N}$ and let $\mathcal{V}(w)$ be the variety of groups defined by the law $w=1$. Define $\mathcal{V}(w^*)$ to be the class of all groups $G$ in which for any infinite subsets $X_1,…
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 $\beta\N$, but contrarily to the other…
In this paper, we consider a variant of the classical algorithmic problem of checking whether a given word $v$ is a subsequence of another word $w$. More precisely, we consider the problem of deciding, given a number $p$ (defining a…
Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen…
We study the asymptotics and fine-scale behavior of quantitative combinatorial measures of infinite words and related dynamical and algebraic structures. We construct infinite recurrent words $w$ whose complexity functions $p_w(n)$ are…
The study of word equations (or the existential theory of equations over free monoids) is a central topic in mathematics and theoretical computer science. The problem of deciding whether a given word equation has a solution was shown to be…
Let $A_q$ be a $q$-letter alphabet and $w$ be a right infinite word on this alphabet. A subword of $w$ is a block of consecutive letters of $w$. The subword complexity function of $w$ assigns to each positive integer $n$ the number $f_w(n)$…