On the optimality of the uniform random strategy

Abstract

Biased Maker‐Breaker games, introduced by Chvátal and Erdős, are central to the field of positional games and have deep connections to the theory of random structures. The main questions are to determine the smallest bias needed by Breaker to ensure that Maker ends up with an independent set in a given hypergraph. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies certain “container‐type” regularity conditions. This will enable us to answer the main question for hypergraph generalizations of the H‐building games studied by Bednarska and Łuczak as well as a generalization of the van der Waerden games introduced by Beck. We find it remarkable that a purely game‐theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, while the analogous questions about sparse random discrete structures are usually quite challenging.

The Maker--Breaker Rado Game on a Random Set of Integers

“…An upper bound on the size of the largest subset of [n] containing no k-distinct solutions to Ax = b is also required. The following is a consequence of Lemma 4.1 in [14] and Theorem 2 in [13].…”

“…Thus both the H-games and (A, b)-games (in most cases) have a threshold bias which is the inverse of the threshold for the random (respective) Ramsey/Rado theorem and the resilience theorems. Kusch, Rué, Spiegel and Szabó [14] in fact show that there is an intimate link between resilience and the threshold bias, which explains the parameters of m(A) and m 2 (H) appearing for both. They refer to this phenomenon as the probabilistic Turán intuition for biased Maker-Breaker games; see Section 6.4 of [14] for more details.…”

“…We call such a game played on a set of integers X the (A, b)-game on X. The class of all (A, b)-games are known as Rado games (introduced in [14]), due to the intimate link with Rado's partition theorem, which will be discussed shortly.…”

“…The biased game result of [14] is the following. In this paper we mainly focus on the case when A satisfies ( * ), though the case where A does not satisfy ( * ) does feature in our first result and also Section 4.…”

“…The main result of this paper is the following. First note that it follows from a supersaturation result (Lemma 4.1 in [14]) that for all pairs (A, b) as stated in Theorem 3, there exist n 0 = n 0 (A, b), δ = δ(A, b) > 0, such that for all integers n ≥ n 0 we have µ(n, A, b) ≤ (1 − δ)n. Thus in particular Theorem 3(i) implies that there exists a positive constant C such that if p > Cn −1/m(A) , then Maker wins the (A, b)-game on [n] p w.h.p. Also, note that if A is a 1 × k matrix with non-zero entries, then it is strictly balanced, and so B(A) = A.…”