Title: Erd\H{o}s-Szekeres Maker-Breaker Games

Abstract. This talk presents new results on Maker-Breaker games arising
from the Erd\H{o}s-Szekeres problem in planar geometry. This classical
problem asks how large a set in general position has to be to ensure the
existence of $n$ points that are the vertices of a convex $n$-gon.
Moreover, Erd\H{o}s further extended this problem by asking what happens
if we also require that this $n$-gon has an empty interior. In a
2-player Maker-Breaker setting, this problem inspires two main games. In
both games, Maker tries to obtain an empty convex $n$-gon, while Breaker
tries to prevent her from doing so. The games differ only in which
points can comprise the winning $n$-gons: in the monochromatic version
the points of both players can make up an $n$-gon, while in the
bichromatic version only Maker's points contribute to such a polygon. We
show that in the monochromatic game, Maker always wins. Even in a biased
game where Breaker is allowed to place $s$ points per round, for any
constant $s \geq 1$, Maker has a winning strategy.  In the bichromatic
setting, Maker still wins whenever Breaker is allowed to place at most
two points per round. Furthermore, we show that Breaker wins when $n=8$
and she is allowed to play 12 or more points per round. We also consider
the two-round bichromatic game (a.k.a.\ the offline version). In this
setting, we show that Maker wins if she has any advantage in the number
of points placed, and we also show that Breaker wins if she can place
twice as many points as Maker. (Joint work with Aleksa D\v{z}uklevski, 
Alexey Pokrovskiy, Tom\'{a}\v{s} Valla and Lander Verlinde.)