Surplus of Graphs and the Lovász Local Lemma

Abstract

The Surplus is a game-theoretic graph parameter. To illustrate it in a special case, suppose that two players, called Maker and Breaker, are playing on an n×n chessboard, and alternately mark previously unmarked little squares. Maker uses (say) mark X and Breaker uses (say) O, exactly like in Tic-Tac-Toe; Maker’s goal is to achieve a large lead in some line, where a “line” means either a row or a column. Let $\frac{n} {2} + \Delta$ denote the maximum number of Xs (“Maker’s mark”) in some line at the end of a play; then the difference $\left( {\frac{n} {2} + \Delta } \right) - \left( {\frac{n} {2} - \Delta } \right) = 2\Delta$ is Maker’s lead; Maker wants to maximize Δ = Δ(n). Since Δ = Δ = Δ(n) can be a half-integer (it happens when n is odd), and it is customary to work with integral graph parameters (like chromatic number), I prefer to call 2Δ = 2Δ(n) the Surplus of the n×n board (and refer to Δ = Δ(n) as the half-surplus). That is, the Surplus is the maximum terminal lead that Maker can always achieve against a perfect opponent. (In other words, Surplus is a game-theoretic one-sided discrepancy concept.)