22 research outputs found
Rainbow Free Colorings and Rainbow Numbers for
An exact r-coloring of a set is a surjective function . A rainbow solution to an equation over is a solution
such that all components are a different color. We prove that every 3-coloring
of with an upper density greater than
contains a rainbow solution to . The rainbow number for an equation in
the set is the smallest integer such that every exact -coloring has
a rainbow solution. We compute the rainbow numbers of for the
equation , where is prime and
Anti-van der Waerden Numbers of Graph Products with Trees
Given a graph , an exact -coloring of is a surjective function
. An arithmetic progression in of length with
common difference is a set of vertices such that
for . An arithmetic progression is rainbow
if all of the vertices are colored distinctly. The fewest number of colors that
guarantees a rainbow arithmetic progression of length three is called the
anti-van der Waerden number of and is denoted . It is known that
. Here we determine exact values for some trees and , determine for some trees
, and determine for some graphs and .Comment: 20 pages, 3 figure
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs