879 research outputs found

    Positional Games

    Full text link
    Positional games are a branch of combinatorics, researching a variety of two-player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs and hypergraphs. It is closely connected to many other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic combinatorics, and to computer science. We survey the basic notions of the field, its approaches and tools, as well as numerous recent advances, standing open problems and promising research directions.Comment: Submitted to Proceedings of the ICM 201

    On the optimality of the uniform random strategy

    Full text link
    The concept of biased Maker-Breaker games, introduced by Chv\'atal and Erd{\H o}s, is a central topic in the field of positional games, with deep connections to the theory of random structures. For any given hypergraph H{\cal H} the main questions is to determine the smallest bias q(H)q({\cal H}) that allows Breaker to force that Maker ends up with an independent set of H{\cal H}. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies a couple of natural `container-type' regularity conditions about the degree of subsets of its vertices. This will enable us to derive a hypergraph generalization of the HH-building games, studied for graphs by Bednarska and {\L}uczak. Furthermore, we investigate the biased version of generalizations of the van der Waerden games introduced by Beck. We refer to these generalizations as Rado games and determine their threshold bias up to constant factors by applying our general criteria. We find it quite remarkable that a purely game theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, when the generalizations to hypergraphs in the analogous setup of sparse random discrete structures are usually quite challenging.Comment: 26 page

    Generating random graphs in biased Maker-Breaker games

    Full text link
    We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for b=o(n)b=o\left(\sqrt{n}\right), Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a (1:b)(1:b) game on E(Kn)E(K_n). As another application, we show that for b=Θ(n/ln⁡n)b=\Theta\left(n/\ln n\right), playing a (1:b)(1:b) game on E(Kn)E(K_n), Maker can build a graph which contains copies of all spanning trees having maximum degree Δ=O(1)\Delta=O(1) with a bare path of linear length (a bare path in a tree TT is a path with all interior vertices of degree exactly two in TT)
    • 

    corecore