In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers n and k, the expression aw([n], k) denotes the smallest number of colors with which the integers {1, . . . , n} can be colored and still guarantee there is a rainbow arithmetic progression of length k. We establish that aw([n], 3) = Θ(log n) and aw([n...

Rapid industrialization and urbanization results in the discharge of large amounts of waste to the environment, which in turn creates more pollution (Senan and Abraham, 2004). Synthetic dyes are extensively used in textile dyeing, paper printing, color photography, pharmaceutical, food, cosmetics and other industries. During textile dyeing, the amount of dye lost in effluent is dependent upon t...

Let εD = v+u √ D be the fundamental unit of Z[ √ D] with Z being the ordinary integers, or maximal order, in the rational field Q. We prove that for any square-free integer D > 1, with D not dividing u, there exists a prime fD such that the relative class number HD(fD) = hf2 DD/hD = 1, where hD is the ideal class number of Z[ √ D] and hf2 DD is the ideal class number of Z[fD √ D], the order of ...

Let R be a subring of the complex numbers and a be a cardinal. A system of linear homogeneous equations with coefficients in R is called a-regular over R if for every a-coloring of R there is a monochromatic solution to that system in distinct variables. Rado in 1943 classified those systems of linear homogeneous equations that are a-regular over R for all positive integers a. For every infinit...

For n ≥ 1, let an count the number of ways one can tile a 1 × n chessboard using 1× 1 square tiles, which come in w colors, and 1× 2 rectangular tiles, which come in t colors. The results for an generalize the Fibonacci numbers and provide generalizations of many of the properties satisfied by the Fibonacci and Lucas numbers. We count the total number of 1× 1 square tiles and 1× 2 rectangular t...

According to the Erdős-Szekeres theorem, every set of n points in the plane contains roughly logn in convex position. We investigate how this bound changes if our point set does not contain a subset that belongs to a fixed order type.

An integer composition of a nonnegative integer n is a tuple (π1, . . . , πk) of nonnegative integers whose sum is n; the πi’s are called the parts of the composition. For fixed number k of parts, the number of f -weighted integer compositions (also called f -colored integer compositions in the literature), in which each part size s may occur in f(s) different colors, is given by the extended b...

Given a set {x1, x2, . . . , xn}, each element of which is colored either red or blue, we must determine an element of the majority color by making equal/not equal color comparisons xu : xv; when n is even, we must report that there is no majority if there are equal numbers of each color. How many such questions are necessary and sufficient? It is easy to obtain an algorithm using at most n−ν(n...

Let G = (V,E) be a graph, let f : V (G)→ N, and let k ≥ 0 be an integer. A list-assignment L of G is a function that assigns to each vertex v of G a set (list) L(v) of colors: usually each color is a positive integer. We say that L is an f -assignment if |L(v)| = f(v) for all v ∈ V , and a k-assignment if |L(v)| = k for all v ∈ V . A coloring ofG is a function φ that assigns a color to each ver...

This study involves in vitro androgenesis of <sp...