# Coloring the Integers with Rainbow Arithmetic Progressions

Friday, March 27, 2015 - 3:30pm - 4:00pm

Keller 3-180

Michael Young (Iowa State University)

A $k$-term arithmetic progression is a sequence of the form $a, a+d, a+2d, ... , a+(k-1)d$, where $a$ and $d$ are nonegative integers. Van der Waerden's Theorem states that given a set of colors there exists an interval $[1,n]$ such that any coloring of the integers, using all the colors, will contain a $k$-term arithmetic progression with each term having the same color. Given a set of colors and $k>0$, actually determining $n$, called a \emph{van der Waerden number} has proven to be a very challenging problem. In this talk, we will discuss some known results about van der Waerden numbers and introduce anti-van der Waerden numbers. An \emph{anti-van der Waerden number} is the number of colors needed to guarantee that any coloring of the interval $[1,n]$ with all the colors must contain a $k$-term arithmetic progression with each term having a distinct color.

MSC Code:

11B25

Keywords: