Related papers: A Coloring Problem for Infinite Words

A simple graph-product type construction shows that for all natural numbers $r \ge q$, there exists an edge-coloring of the complete graph on $2^r$ vertices using $r$ colors where the graph consisting of the union of arbitrary $q$ color…

We study the number of monochromatic solutions to linear equations in a $2$-coloring of $\{1,\ldots,n\}$. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any $2$-coloring of…

In this paper, we provide a new characterization of uniformly recurrent words with finite defect based on a relation between the palindromic and factor complexity. Furthermore, we introduce a class of morphisms P_ret closed under…

We show that various analogs of Hindman's Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1: There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that for every…

We show that in every two-colouring of the edges of the complete graph $K_N$ there is a monochromatic $K_k$ which can be extended in at least $(1 + o_k(1))2^{-k}N$ ways to a monochromatic $K_{k+1}$. This result is asymptotically best…

A classical question in combinatorial number theory asks whether an equation has a solution inside a particular subset of its domain. The Rado's Theorem gives a set of necessary and sufficient conditions for a systems of linear equations to…

Let $f(n,r)$ denote the maximum number of colourings of $A \subseteq \lbrace 1,\ldots,n\rbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $\lbrace x,y,z\rbrace$ such that $x+y=z$. We show that $f(n,2)…

An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey Theorem on uncountable cardinals asserting…

Given a probability space $(X, {\cal B}, m)$, measure preserving transformations $g_1, \dots , g_k$ of $X$, and a colour set $C$, a colouring rule is a way to colour the space with $C$ such that the colours allowed for a point $x$ are…

We prove that for any $r\in \mathbb{N}$, there exists a constant $C_r$ such that the following is true. Let $\mathcal{F}=\{F_1,F_2,\dots\}$ be an infinite sequence of bipartite graphs such that $|V(F_i)|=i$ and $\Delta(F_i)\leq \Delta$ hold…

In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions that homogenize colorings over…

Given a graph $G$, its Ramsey number $r(G)$ is the minimum $N$ so that every two-coloring of $E(K_N)$ contains a monochromatic copy of $G$. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from $G$, the…

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…

Suppose that $\mathbb{N}$ is $2$-coloured. Then there are infinitely many monochromatic solutions to $x + y = z^2$. On the other hand, there is a $3$-colouring of $\mathbb{N}$ with only finitely many monochromatic solutions to this…

We prove that every (possibly infinite) graph of degree at most $d$ has a 4-dependent random proper $4^{d(d+1)/2}$-coloring, and one can construct it as a finitary factor of iid. For unimodular transitive (or unimodular random) graphs we…

The Lov\'asz Local Lemma is a powerful probabilistic technique for proving the existence of combinatorial objects. It is especially useful for colouring graphs and hypergraphs with bounded maximum degree. This paper presents a general…

The stable Ramsey's theorem for pairs has been the subject of numerous investigations in mathematical logic. We introduce a weaker form of it by restricting from the class of all stable colorings to subclasses of it that are non-null in a…

Given two combinatorial notions $\mathsf{P}_0$ and $\mathsf{P}_1$, can we encode $\mathsf{P}_0$ via $\mathsf{P}_1$. In this talk we address the question where $\mathsf{P}_0$ is 3-coloring of integers and $\mathsf{P}_1$ is product of…

Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points of distance one. For every natural number $n$, we prove the existence of a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of…

A \emph{unichord} in a graph is an edge that is the unique chord of a cycle. A \emph{square} is an induced cycle on four vertices. A graph is \emph{unichord-free} if none of its edges is a unichord. We give a slight restatement of a known…