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

TL;DR

This paper introduces the Erd ext{"o}s-Szekeres on-line number, an online variant of a classical combinatorial problem, analyzing its bounds and behavior when points are chosen sequentially.

Contribution

It defines the on-line version of the Erd ext{"o}s-Szekeres problem, establishes bounds, and determines the value for specific parameters, advancing understanding of online combinatorial geometry.

Findings

ext{ESO}(m,k) < (m-1)(k-1)+1$ for $m,k ext{ at least } 3$

Provided a general lower bound for ext{ESO}(m,k)

Determined ext{ESO}(m,3) up to an additive constant

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.