Erd\H{o}s-Szekeres On-Line
Kirk Boyer, Lauren M. Nelsen, Luke L. Nelsen, Florian Pfender,, Elizabeth Reiland, Ryan Solava

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