On K-AP And K-Avoid-AP Van Der Waerden Game

Abstract

Let $w(k;2)$ be the van der Waerden number such that for every 2-colouring of $[1,w(k;2)]$ there is a monochromatic $k$-term arithmetic progression (AP). Consider the following two 2-players games: $k$-AP game and $k$-AVOID-AP game. These are two different games between two players, Player 1 and Player 2, on a sequence of integers $[1,n]$ where $n\in \mathbb{Z}^+$. Each player's aim is to obtain or avoid forming a monochromatic $k$-term arithmetic progression. The player who first obtains a monochromatic $k$-term arithmetic progression wins or loses, thus ending the game. In this paper, we investigate these two games and propose a new parameter: the minimum number of turns $\hat{w}_n(k)$ (and $\tilde{w}_n(k)$) needed for any of the player to win in $k$-AP (and $k$-AVOID-AP game respectively). We propose the winning strategies for Player 1 and Player 2 and hence show that $\hat{w}_n(3)=5, \hat{w}_n(4)=7$ and $\tilde{w}_n(3)=9$. We also have shown that in a $k$-AVOID-AP game on $[1,n]$, where $n$ is sufficiently large, Player 2 always has a winning strategy if $n$ is even.