Related papers: Restricted Color $n$-color Compositions
In this article, we consider systems of linear congruences in several variables and obtain necessary and sufficient conditions as well as explicit expressions for the number of solutions subject to certain restriction conditions. These…
For which values of $n$ can we color the positive integers with precisely $n$ colors in such a way that for any $a$, the numbers $a,2a,\dots,na$ all get different colors? Pach posed the question around 2008-9. Particular cases appeared in…
Motivated by a new way of visualizing hypergraphs, we study the following problem. Consider a rectangular grid and a set of colors $\chi$. Each cell $s$ in the grid is assigned a subset of colors $\chi_s \subseteq \chi$ and should be…
A barred preferential arrangement is a preferential arrangement onto which a number of identical bars are inserted in between the blocks of the preferential arrangement. In this study we examine combinatorial properties of barred…
The problem of enumerating meanders -- pairs of simple plane curves with transverse intersections -- was formulated about forty years ago and is still far from solved. Recently, it was discovered that meanders admit a factorization into…
The Hales-Jewett theorem states that for any $m$ and $r$ there exists an $n$ such that any $r$-colouring of the elements of $[m]^n$ contains a monochromatic combinatorial line. We study the structure of the wildcard set $S \subseteq [n]$…
Colouring problems arising from group-based constructions provide a natural link between combinatorics and algebra, particularly in the study of Cayley graphs and Latin squares. We introduce the notion of colouring bijections of finite…
Coloring graphs is an important algorithmic problem in combinatorics with many applications in computer science. In this paper we study coloring tournaments. A chromatic number of a random tournament is of order $\Omega(\frac{n}{\log(n)})$.…
The {\em d} -- dimensional $n$ -- colour lattice ${\bf \Bbb{L}}^d$ with modular sublattices are studied, when the only one crystallographic type of sublattices does exist and the only one of the colours occupies a sublattice, which is still…
We provide a general and unified combinatorial framework for a number of colored partition identities, which include the five, recently proved analytically by B. Berndt, that correspond to the exceptional modular equations of prime degree…
The idea that those different from you are "unfriendly" is captured in the definition of unfriendly 2-colorings in graph theory in a paper by Aharoni, Milner and Prikry, where they prove that every finite graph has an unfriendly coloring.…
Information about the illuminant color is well contained in both achromatic regions and the specular components of highlight regions. In this paper, we propose a novel way to achieve color constancy by exploiting such clues. The key to our…
We study some aspects of divisionally free arrangements which were recently introduced by Abe. Crucially, Terao's conjecture on the combinatorial nature of freeness holds within this class. We show that while it is compatible with products,…
In this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L \geq 5 for p \geq 11 and mincol_p L \geq 6…
A structure $\mathcal{A}=\left(A;E_i\right)_{i\in n}$ where each $E_i$ is an equivalence relation on $A$ is called an $n$-grid if any two equivalence classes coming from distinct $E_i$'s intersect in a finite set. A function $\chi: A \to n$…
Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with colored cubes and bricks of $(2\times2\times n)$-board in three…
We take a unified approach to central limit theorems for a class of irreducible urn models with constant replacement matrix. Depending on the eigenvalue, we consider appropriate linear combinations of the number of balls of different…
In 2001, Bhargava proved a composition law for $2 \times 2 \times 2$ integer cubes, which generalized Gauss composition of integral binary quadratic forms. Furthermore, he derived four new composition laws defined on the following spaces:…
Given an equation, the integers $[n] = \{1, 2, \dots, n\}$ as inputs, and the colors red and blue, how can we color $[n]$ in order to minimize the number of monochromatic solutions to the equation, and what is the minimum? The answer is…
This paper is intended to provide concrete examples of concepts discussed elsewhere in this volume, especially splittings of groups and non-positively curved cube complexes but also other things. The idea of the construction (configuration…