Arithmetic progressions in partially ordered sets

Van der Waerden's arithmetic sequence theorem—in particular, the ‘density version’ of Szemerédi—is generalized to partially ordered sets in the following manner. Let w and t be fixed positive integers and ε>0. Then for every sufficiently large partially ordered set P of width at most w, every subset S of P satisfying |S|≥ε|P| contains a chain a1, a2,..., a1 such that the cardinality of the interval [ai, ai+1] in P is the same for each i.