Endgame problems of Sim-like graph Ramsey avoidance games are PSPACE-complete

AbstractIn Sim, two players compete on a complete graph of six vertices (K6). The players alternate in coloring one as yet uncolored edge using their color. The player who first completes a monochromatic triangle (K3) loses. Replacing K6 and K3 by arbitrary graphs generalizes Sim to graph Ramsey avoidance games. Given an endgame position in these games, the problem of deciding whether the player who moves next has a winning strategy is shown to be PSPACE-complete. It can be reduced from the problem of whether the first player has a winning strategy in the game Gpos(POSCNF) (Schaefer, J. Comput. System Sci. 16 (2) (1978) 185–225). The following game variants are also shown to have PSPACE-complete endgame problems: (1) completing a monochromatic subgraph isomorphic to A is forbidden and the player who is first unable to move loses, (2) both players are allowed to color one or more edges in each move, (3) more than two players take part in the game, and (4) each player has to avoid a separate graph. In all results, the graphs to be avoided can be restricted to the bowtie graph (⋈, i.e., two triangles with one common vertex)

6-Uniform Maker-Breaker Game Is PSPACE-Complete

In a STOC 1976 paper, Schaefer proved that it is PSPACE-complete to determine the winner of the so-called Maker-Breaker game on a given set system, even when every set has size at most 11. Since then, there has been no improvement on this result. We prove that the game remains PSPACE-complete even when every set has size 6