Van der waerden and ramsey type games

Abstract

Let us consider the following 2-player game, called van 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 an n-term arithmetic progression. Let W*(n) be the least integer N so that the first player has a winning strategy. By the Ramsey game on k-tuples we shall mean a 2-player game where the players alternately pick previously unpicked elements of the complete k-uniform hypergraph of N vertices KNk, and the first player wins if he has selected all k-tuples of an n-set. Let Rk*(n) be the least integer N so that the first player has a winning strategy. We prove (W* (n))1/n → 2, R2*(n) <2nk /k! for k ≧3. © 1981 Akadémiai Kiadó.