Erd\H{o}s-Szekeres On-Line

Abstract

In 1935, Erd\H{o}s and Szekeres proved that $(m-1)(k-1)+1$ is the minimum number of points in the plane which definitely contain an increasing subset of $m$ points or a decreasing subset of $k$ points (as ordered by their $x$-coordinates). We consider their result from an on-line game perspective: Let points be determined one by one by player A first determining the $x$-coordinate and then player B determining the $y$-coordinate. What is the minimum number of points such that player A can force an increasing subset of $m$ points or a decreasing subset of $k$ points? We introduce this as the Erd\H{o}s-Szekeres on-line number and denote it by $\text{ESO}(m,k)$. We observe that $\text{ESO}(m,k) < (m-1)(k-1)+1$ for $m,k \ge 3$, provide a general lower bound for $\text{ESO}(m,k)$, and determine $\text{ESO}(m,3)$ up to an additive constant.