Variations on a game

Abstract

This survey paper contains several nice results of the authors on special positional games called amoeba Two players, Maker and Breaker, move alternately with Maker going first. In each turn they occupy previously unoccupied vertices of a hypergraph. In the game Maker's aim is to occupy all vertices of some edge of the hypergraph before Breaker can do it. In the game Braker's aim is to prevent Maker from achieving her goal without looking at his own pieces. For example, in the Van der Waerden game the winning sets are the $n$ term arithmetic progressions for some fixed $n>1$. We discuss this, and several other games, when can Maker and when can Breaker win. We show that the weak games have compactness property, and we give counterexamples for strong games. We also consider infinite games where the players keep on taking moves until there are any unoccupied vertex in an infinite grap.