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The partition X = A1 ∪ A2 ∪ · · · ∪ Ar is often called an r-coloring of X, where A1 , , Ar are the color classes. For example, if X is the 7-point Moser-graph in the plane, the “goal set” S consists of two points a unit distance apart, and r = 3. g. Chapter 11 in the Handbook of Combinatorics). Here we study a game-theoretic version, and prove a very general result in a surprisingly elementary way. The game-theoretic version goes as follows: there are two players, called Maker and Breaker, who alternately select new points from some Euclidean space Rd .

Unlike these two very successful theories, we still do not have a really satisfying quantitative theory of games of pure skill with complete information, or as they are usually called nowadays: Combinatorial Games. Using technical terms, Combinatorial Games are 2-player zero-sum games, mostly finite, with complete information and no chance moves, and the payoff function has three values ±1 0 as the first player wins or loses the play, or it ends in a draw. e. ” Note that Graph Theory and Combinatorial Game Theory face the very same challenge: combinatorial chaos.

In either case the first player occupies X2 , X3 , X4 , and completes a winning triplet. Of course, this way the opponent’s winning triplet (O1 O2 O3 if the second player plays rationally) comes first; notice that here we changed the rule and assumed that the players do not quit playing even after some winning set is completely occupied by either player, they just keep playing till the whole board is completed. We refer to this as the Full Play Convention. Case 1 Case 2 O1 O1 X1 X1 X2 X3 X4 X2 X4 X3 Occupying a whole winning set, but not necessarily first, is what we call a Weak Win.

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