Fractional dimension of partial orders

Given a partially ordered setP=(X, ), a collection of linear extensions {L₁,L₂,...,L_r} is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someL_i (andy<x in someL_j). For a positive integerk, we call a multiset {L₁,L₂,...,L_t} ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theL_i's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask. We prove various results about the fractional dimension of a poset.Research supported in part by the Office of Naval Research.