## The infinite random order of dimension k.(English)Zbl 0589.06001

Fix an integer $$\kappa\geq 1$$ and let $$I^{\kappa}$$ be the unit hypercube in Euclidean $$\kappa$$-space, endowed with the ordinary product order. If n points are chosen randomly and independently from the uniform probability distribution on $$I^{\kappa}$$, the resulting ordered set is called the random order $$P_ n^{\kappa}$$. The author discusses the problem, whether ”the 0-1 law” holds for random orders, i.e. a theorem saying that for any first order sentence S in the language of ordered sets the $$\lim_{n\to \infty}P(S$$ holds in $$P_ n^{\kappa})$$ is either 0 or 1. The conjecture is made that the answer is yes in general, no for fixed or bounded dimension.
Reviewer: V.N.Salij

### MSC:

 06A06 Partial orders, general 03C90 Nonclassical models (Boolean-valued, sheaf, etc.) 60C05 Combinatorial probability