Connectedness and diameter for random orders of fixed dimension (Q1068112)

Choose n points independently from the uniform distribution in \([0,1]^ k\) and define \(x\leq y\) if \(x_ i\leq y_ i\) for each i, \(1\leq i\leq k\). The obtained partial order \(P_ k(n)\) is said to be connected if the comparability graph \(G_ k(n)\) of \(P_ k(n)\) is connected. The diameter diam \(P_ k(n)\) is the least j such that any two vertices of \(G_ n(k)\) are connected by a path of length at most j. Theorem. Fix \(k>1\), then with probability approaching 1 as \(n\to \infty\), \(P_ k(n)\) is connected and diam \(P_ k(n)=3\).