On increasing sequences formed by points from a random finite subset of a hypercube

Abstract

Consider $S$, a set of $n$ points chosen uniformly at random and independently from the unit hypercube of dimension $t>2$. Order $S$ by using the Cartesian product of the $t$ standard orders of $[0,1]$. We determine a constant $\bar x(t)<e$ such that, with probability $\ge 1-\exp(-\Theta(\eps)n^{1/t})$, cardinality of a largest subset of comparable points is at most $(\bar x(t)+\eps)n^{1/t}$. The bound $\bar x(t)$ complements an explicit lower bound obtained by Bollob\'as and Winkler in 1982. Furthermore, we use Dilworth's theorem on partitions of a set into chains to prove that the cardinality of a largest antichain, i. e. a largest subset of incomparable points, is at least $(1-\eps) (n/e)^{1-1/t}$ with probability exponentially close to $1$.