Longest k-monotone chains

TL;DR

This paper investigates the maximum size of k-monotone chains in random point sets within the unit square, providing asymptotic estimates and concentration results that generalize known cases for increasing sequences and convex chains.

Contribution

It introduces a general framework for analyzing k-monotone chains, extending previous results for specific cases to a broader class of higher-order convexity properties.

Findings

Asymptotic estimates for the maximal number of k-monotone points

Strong concentration estimates for these maximal chains

A unified framework encompassing previous special cases

Abstract

We study higher order convexity properties of random point sets in the unit square. Given $n$ uniform i.i.d random points, we derive asymptotic estimates for the maximal number of them which are in $k$-monotone position, subject to mild boundary conditions. Besides determining the order of magnitude of the expectation, we also prove strong concentration estimates. We provide a general framework that includes the previously studied cases of $k=1$ (longest increasing sequences) and $k=2$ (longest convex chains).