Random Strategies are Nearly Optimal for Generalized van der Waerden Games

Abstract

In a (1 : q) Maker-Breaker game, one of the central questions is to find (or at least estimate) the maximal value of q that allows Maker to win the game. Based on the ideas of Bednarska and Luczak [1], who studied biased H-games, we prove general winning criteria for Maker and Breaker and a hypergraph generalization of their result. Furthermore, we study the biased version of a strong generalization of the van der Waerden games introduced by Beck [2] and apply our criteria to determine the threshold bias of these games up to constant factor. As in the result of [1], the random strategy for Maker is again the best known strategy.

A Note on Sparse Supersaturation and Extremal Results for Linear Homogeneous Systems

“…The next lemma is crucial and establishes that a lack of non-trivial solutions to a subsystem of A also implies a lack of non-trivial solutions to the full system. A proof of this as well as the previous statement can be found in Kusch et al [21]. Note that this was previously proven by Rödl and Ruciński for proper solutions [6].…”

“…If A p is not partition regular, then [n] → s A and therefore trivially lim n→∞ P Ap) and therefore again by Theorem 1.1 we have lim n→∞ P [n] p → s A p = 0 for p = p(n) ≤ c n −1/m1(A) . The desired statement follows due to Equation (21).…”

“…Lemma 2.6 (Kusch et al [21]). For any r, m ∈ N, abundant matrix A ∈ M r×m (Z) and selection of column indices Q ⊆ [m] the following holds.…”

A new Bound for the Maker-Breaker Triangle Game

“…Bednarska and Luczak [6] used it to prove the following fundamental result: For a fixed graph G consider the (G, n, q)-game, in which Maker has to build a copy of G. There exist constants c 0 , C 0 such that the game is a Maker's win for q ≤ c 0 n 1/m(G) and a Breaker's win for q ≥ C 0 n 1/m (G) , where m(G) := max e(H)−1 v(H)−1 : H ⊆ G, v(H) ≥ 3 . This result recently was further generalized for hypergraphs by Kusch et al [8]. Bednarska and Luczak also conjectured that c 0 and C 0 can be chosen arbitrarily close to each other.…”