Arithmetic progression game

The arithmetic progression game is a positional game where two players alternately pick numbers, trying to occupy a complete arithmetic progression of a given size.

The game is parameterized by two integers n > k. The game-board is the set {1,...,n}. The winning-sets are all the arithmetic progressions of length k. In a Maker-Breaker game variant, the first player (Maker) wins by occupying a k-length arithmetic progression, otherwise the second player (Breaker) wins.

The game is also called the van der Waerden game,{{Cite journal|last=Beck|first=József|date=1981|title=Van der Waerden and Ramsey type games|journal=Combinatorica|language=en|volume=1|issue=2|pages=103–116|doi=10.1007/bf02579267|issn=0209-9683}} named after Van der Waerden's theorem. It says that, for any k, there exists some integer W(2,k) such that, if the integers {1, ..., W(2,k)} are partitioned arbitrarily into two sets, then at least one set contains an arithmetic progression of length k. This means that, if $n \geq W(2,k)$ , then Maker has a winning strategy.

Unfortunately, this claim is not constructive - it does not show a specific strategy for Maker. Moreover, the current upper bound for W(2,k) is extremely large (the currently known bounds are: $2^{k}/k^\varepsilon < W(2,k) < 2^{2^{2^{2^{2^{k+9}}}}}$ for every $\varepsilon>0$ ).

Let W*(2,k) be the smallest integer such that Maker has a winning strategy. Beck proves that $2^{k-7k^{7/8}} < W^*(2,k) < k^3 2^{k-4}$ . In particular, if $k^3 2^{k-4} < n$ , then the game is Maker's win (even though it is much smaller than the number that guarantees no-draw).

References

Category:Positional games