879 research outputs found
Positional Games
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
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 the
main questions is to determine the smallest bias that allows
Breaker to force that Maker ends up with an independent set of . 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 -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
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
, Maker can build a pancyclic graph (that is, a graph
that contains cycles of every possible length) while playing a game on
. As another application, we show that for , playing a game on , Maker can build a graph which
contains copies of all spanning trees having maximum degree with
a bare path of linear length (a bare path in a tree is a path with all
interior vertices of degree exactly two in )
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