Related papers: Finitely dependent cycle coloring

We use the Mallows permutation model to construct a new family of stationary finitely dependent proper colorings of the integers. We prove that these colorings can be expressed as finitary factors of i.i.d. processes with finite mean coding…

Holroyd and Liggett recently proved the existence of a stationary 1-dependent 4-coloring of the integers, the first stationary k-dependent q-coloring for any k and q. That proof specifies a consistent family of finite-dimensional…

The existence of stationary finitely dependent processes on combinatorial models like $\mathbb Z^d$ subshifts can be quite mysterious. For instance, Holroyd and Liggett constructed such processes on proper $4$-colorings of $\mathbb Z^d$ for…

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…

A $q$-coloring of $\mathbb Z$ is a random process assigning one of $q$ colors to each integer in such a way that consecutive integers receive distinct colors. A process is $k$-dependent if any two sets of integers separated by a distance…

We prove that proper coloring distinguishes between block-factors and finitely dependent stationary processes. A stochastic process is finitely dependent if variables at sufficiently well-separated locations are independent; it is a…

We prove the existence of a finitely dependent proper colouring of the integer lattice Z^d that is fully isometry-invariant in law, for all dimensions d. Previously this was known only for d=1, while only translation-invariant examples were…

We use the combinatorial properties of central sets to prove a result about the existence of exponential monochromatic patterns, in the style of Hindman's Finite Sums Theorem. More precisely, we prove that for every finite coloring of the…

We show that any finitely dependent invariant process on a transitive amenable graph is a finitary factor of an i.i.d. process. With an additional assumption on the geometry of the graph, namely that no two balls with different centers are…

This is a treatise on finite point configurations spanning a fixed volume to be found in a single color-class of an arbitrary finite (measurable) coloring of the Euclidean space $\mathbb{R}^n$, or in a single large measurable subset…

In a recent paper by the same authors, we constructed a stationary 1-dependent 4-coloring of the integers that is invariant under permutations of the colors. This was the first stationary k-dependent q-coloring for any k and q. When the…

In this paper we introduce mixed coloured permutation, permutations with certain coloured cycles, and study the enumerative properties of these combinatorial objects. We derive the generating function, closed forms, recursions and…

We describe how to construct a spanning set of linearly-independent, automatically orthogonal colour tensors for scattering amplitudes involving coloured particles transforming under arbitrary representations of any gauge theory, sufficient…

It is consistent for every (1 <= n< omega) that (2^omega = omega_n) and there is a function (F:[omega_n]^{< omega}-> omega) such that every finite set can be written at most (2^n-1) ways as the union of two distinct monocolored sets. If GCH…

We study compositions whose parts are colored by subsequences of the Fibonacci numbers. We give explicit bijections between Fibonacci colored compositions and several combinatorial objects, including certain restricted ternary and…

We discuss and prove a number of results for calculating characteristic cycles, or graded, enriched characteristic cycles. We concentrate particularly on results related to hypersurfaces.

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e.,…

We consider a robust variant of Dirac-type problems in $k$-uniform hypergraphs. For instance, we prove that if $H$ is a $k$-uniform hypergraph with minimum codegree at least $(1/2 + \gamma )n$, $\gamma >0$, and $n$ is sufficiently large,…

This paper considers synchronous discrete-time dynamical systems on graphs based on the threshold model. It is well known that after a finite number of rounds these systems either reach a fixed point or enter a 2-cycle. The problem of…

We consider the existence of patterned Hamilton cycles in randomly colored random graphs. Given a string $\Pi$ over a set of colors $\{1,2,\ldots,r\}$, we say that a Hamilton cycle is $\Pi$-colored if the pattern repeats at intervals of…