Related papers: Coloring finite subsets of uncountable sets

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 show that for every coloring of the rationals into finitely many colors, one of the colors contains a set of the form $\{x,y,xy,x+y\}$ for some nonzero $x$ and $y$.

We conclude an investigation of Abrishami, Esperet, Giocanti, Hamman, Knappe and M\"oller studying the existence of periodic colourings of locally finite graphs. A colouring of a graph $\Gamma$ is periodic if the resulting coloured graph…

We investigate proper $(a:b)$-fractional colorings of $n$-uniform hypergraphs, which generalize traditional integer colorings of graphs. Each vertex is assigned $b$ distinct colors from a set of $a$ colors, and an edge is properly colored…

We define a $P$-compelling coloring as a proper coloring of the vertices of a graph such that every subset consisting of one vertex of each color has property $P$. The $P$-compelling chromatic number is the minimum number of colors in such…

Let $\gamma_n$ be the permutation on $n$ symbols defined by $\gamma_n = (1\ 2\...\ n)$. We are interested in an enumerative problem on colored permutations, that is permutations $\beta$ of $n$ in which the numbers from 1 to $n$ are colored…

Given two graphs $F$ and $G$, an $F$-WORM coloring of $G$ is an assignment of colors to its vertices in such a way that no $F$-subgraph of $G$ is monochromatic or rainbow. If $G$ has at least one such coloring, then it is called $F$-WORM…

In an improper colouring an edge $uv$ for which, $c(u)=c(v)$ is called a \emph{bad edge}. The notion of the \emph{chromatic completion number} of a graph $G$ denoted by $\zeta(G),$ is the maximum number of edges over all chromatic…

In this paper, we consider infinite words that arise as fixed points of primitive substitutions on a finite alphabet and finite colorings of their factors. Any such infinite word exhibits a "hierarchal structure" that will allow us to…

We colour every point x of a probability space X according to the colours of a finite list x_1, ...., x_k of points such that each of the x_i, as a function of x, is a measure preserving transformation. We ask two questions about a…

A Star Coloring of a graph G is a proper vertex coloring such that every path on four vertices uses at least three distinct colors. The minimum number of colors required for such a star coloring of G is called star chromatic number, denoted…

The smallest integer $k$ needed for the assignment of colors to the elements so that the coloring is proper (vertices and edges) is called the total chromatic number of a graph. Vizing and Behzed conjectured that the total coloring can be…

For fixed integers p and q, let f(n,p,q) denote the minimum number of colors needed to color all of the edges of the complete graph K_n such that no clique of p vertices spans fewer than q distinct colors. A construction is given which…

Given a system $(G_1, \ldots ,G_m)$ of graphs on the same vertex set $V$, a cooperative coloring is a choice of vertex sets $I_1, \ldots ,I_m$, such that $I_j$ is independent in $G_j$ and $\bigcup_{j=1}^{m}I_j = V$. For a class…

Let $G$ be a multigraph and $L\,:\,E(G) \to 2^\mathbb{N}$ be a list assignment on the edges of $G$. Suppose additionally, for every vertex $x$, the edges incident to $x$ have at least $f(x)$ colors in common. We consider a variant of local…

An edge colouring of a graph is called distinguishing if there is no non-trivial automorphism which preserves it. We prove that every at most countable, finite or infinite, connected regular graph of order at least $7$ admits a…

An alternating colouring function is defined on strings over the alphabet $\{0, 1\}$. It divides the strings in colourable and non-colourable ones. The points in the subshift of finite type defined by forbidding all non-colourable strings…

We prove that for any coloring of the naturals using two colors there are monochromatic sets of the form $\{x,y,xy,x+iy:i\leq k\}$ and $\{x,y,x^y,xy^i:i\leq k\}$ for any $k$.

A proper edge $t$-coloring of a graph is a coloring of its edges with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the same color. For any integer $n\geq 3$, all possible values of $t$ are found, for…

A $k$-uniform tight cycle is a $k$-graph with a cyclic order of its vertices such that every $k$ consecutive vertices from an edge. We show that for $k\geq 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains $k$…