On the set coincidence game

In the Set Coincidence Game G(V, W), two players alternately choose elements not previously chosen from a finite, nonempty set V, and W is a given family of nonempty subsets of V (the ‘winning sets’). The winner is that player who first adds an element to the set of ‘chosen’ elements 5, so that S E W. This game is closely related to and generalizes Ringeisen’s Isolation Game on graphs. We develop the theory of G(V, W), present and support a conjecture about the structure of minimal forced wins, and then prove a weakened form (the Weak Filter Theorem). It is hoped that the indicated themes about optima1 design of forced wins will prove of interest for a variety of combinatorial games.