The longest chain among random points in Euclidean space

Author:

Bollobás Béla,Winkler Peter

Abstract

Let n n random points be chosen independently from the uniform distribution on the unit k k -cube [ 0 , 1 ] k {[0,1]^k} . Order the points coordinate-wise and let H k ( n ) {{\mathbf {H}}_k}\left ( n \right ) be the cardinality of the largest chain in the resulting partially ordered set. We show that there are constants c 1 , c 2 , {c_1},{c_2}, \ldots such that c k > e , lim k c k = e {c_k} > e,\;{\lim _{k \to \infty }}{c_k} = e , and lim n H k ( n ) / n 1 / k = c k {\lim _{n \to \infty }}{{\mathbf {H}}_k}\left ( n \right )/{n^{1/k}} = {c_k} in probability. This generalizes results of Hammersley, Kingman and others on Ulam’s ascending subsequence problem, and settles a conjecture of Steele.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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