Directed graphs as unions of partial orders

The index of an irreflexive binary relation R is the smallest cardinal number σ(R) such that R equals the union of σ(R) partial orders. With s(n) the largest index for an R defined on n points, it is shown that s(n)/log₂ n→1 as n →∞. The index function is examined for symmetric R’s and almost transitive R’s, and a characterization for σ(R)≦2 is presented. It is shown also that inf{n:s(n)>3}≦13, but the exact value of inf {n:s(n)>3} is presently unknown.