On the Computational Complexity and Strategies of Online Ramsey Theory

An online Ramsey game (G, H) is a game between Builder and Painter, alternating in turns. In each round Builder draws an edge and Painter colors it either red or blue. Builder wins if after some round there is a monochromatic copy of the graph H, otherwise Painter is the winner. The rule for Builder is that after each his move the resulting graph must belong to the class of graphs G. In this abstract we investigate the computational complexity of the related decision problem and we show that it is PSPACE-complete. Moreover, we study a generalization of online Ramsey game for hypergraphs and we provide a result showing that Builder wins the online Ramsey game for the case when G and H are 3-uniform hyperforests and H is 1-degenerate.