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

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 $\ge 1-\exp (-\mathrm{\Theta }(\text{eps})n^{1\mathrm{/}t})$, cardinality of a largest subset of comparable points is at most $(\bar{x}(t)+\text{eps})n^{1\mathrm{/}t}$. The bound $\bar{x}(t)$ complements an explicit lower bound obtained by Bollobás 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 $(1-\text{eps})(n\mathrm{/}e{)}^{1-1\mathrm{/}t}$ with probability exponentially close to $1$.