An exploration of anti-van der Waerden numbers
In this paper results of the anti-van der Waerden number of various mathematical objects are discussed. The anti-van der Waerden number of a mathematical object G, denoted by aw(G,k), is the smallest r such that every exact r-coloring of G contains a rainbow k-term arithmetic progression. In this paper, results on the anti-van der Waerden number of the integers, groups such as the integers modulo n, and graphs are given. A connection between the Ramsey number of paths and the anti-van der Waerden number of graphs is established. The anti-van der Waerden number of [m]X[n] is explored. Finally, connections between anti-van der Waerden numbers, rainbow numbers, and anti-Schur numbers are discussed.