Making vs. Avoinding in Positional Games

by Tibor Szab\'o

A {\em positional game} consists of a set $X$, the "board", and a
family of subsets ${\cal F}\subseteq 2^X$, the set of "winning sets".
During the game two players alternately claim elements of the board.
In a Maker/Breaker-type game, the player called Maker wins if he
succeeds in fully claiming a winning set, while Breaker wins if he can
prevent Maker from doing so. In an Avoider/Enforcer-type game the
player called Avoider wins the game if he does {\em not} occupy any
winning set fully, otherwise Enforcer wins.
Classical positional games are often played on the edges of the
complete graph, and ${\cal F}$ usually expresses some nice graph
theoretic property, like containing a Hamiltonian cycle or being connected.
In the talk we survey our current knowledge about Maker/Breaker- and
Avoider/Enforcer-type positional games, discuss their relations to
each other and to the theory of random graphs.
Based on joint works with Dan Hefetz, Michael Krivelevich and  Milos
Stojakovic.