Van der waerden and ramsey type games

Abstract

Let us consider the following 2-player game, calledvan der Waerden game. The players alternately pick previously unpicked integers of the interval {1, 2, ...,N}. The first player wins if he has selected all members of ann-term arithmetic progression. LetW*(n) be the least integerN so that the first player has a winning strategy.

By theRamsey game on k-tuples we shall mean a 2-player game where the players alternately pick previously unpicked elements of the completek-uniform hypergraph ofN verticesK kN , and the first player wins if he has selected allk-tuples of ann-set. LetR k*(n) be the least integerN so that the first player has a winning strategy.

We prove (W* (n))1/n → 2,R 2*(n)<(2+ε)n andR * k n<2nk/k! fork ≧3.

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Beck, J. Van der waerden and ramsey type games. Combinatorica 1, 103–116 (1981). https://doi.org/10.1007/BF02579267

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