390 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
Minors in expanding graphs
Extending several previous results we obtained nearly tight estimates on the
maximum size of a clique-minor in various classes of expanding graphs. These
results can be used to show that graphs without short cycles and other H-free
graphs contain large clique-minors, resolving some open questions in this area
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