One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map (a Tukey quotient) φ : P → Q carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients of each other are said to be Tukey equivalent. Let Dc be the partially ordered set of Tukey equivalence classes of directed sets of size ≤ c. It is shown that Dc contains an antichain of size 2 c , and so has size 2 c . The elements of the antichain are of the form K(M ), the set of compact subsets of a separable metrizable space M , ordered by inclusion. The order structure of such K(M )'s under Tukey quotients is investigated. Relative Tukey quotients are introduced. Applications are given to function spaces and to the complexity of weakly countably determined Banach spaces and Gul'ko compacta. §1. Introduction. If there is a map φ : P → Q, where P and Q are partial orders, such that φ maps cofinal sets of P to cofinal sets of Q, then Q is said to be a Tukey quotient of P, denoted P ≥ T Q, and φ is called a Tukey quotient. Two partial orders, P and Q, which are mutual Tukey quotients of each other, P ≥ T Q and Q ≥ T P, are said to be Tukey equivalent, abbreviated P = T Q. Introduced to study Moore-Smith convergence in topology [19,26], Tukey quotients and equivalence are fundamental notions of order theory, and are being actively investigated, especially in connection with partial orders arising naturally in analysis and topology [6-10, 14, 18, 20-23].A partially ordered set is directed if every two elements have an upper bound. For a cardinal κ, let D κ be the collection of Tukey equivalence classes of directed sets of size ≤ κ. Note that D κ is partially ordered by Tukey quotients. A key contribution to the theory of Tukey equivalence of directed sets was made by Todorcevic in [25]. He showed that consistently there are just five members of D 1 (namely, 1, , 1 , × 1 and [ 1 ] < ). Todorcevic also established, in ZFC, that for all regular κ, the collection D κ ℵ 0 has size at least 2 κ , because it contains an antichain of size 2 κ . Under CH, then, there are 2 c -many directed sets of size 1 . This pair of results gives a rather dramatic answer to Isbell's question [12] as to the size of D 1 , 'maybe 5, maybe 2 c , it depends on your set theory'.Left unresolved is the size of D c . Evidently |D c | ≤ 2 c . From Todorcevic's second result it is at least 2 κ for any regular κ ≤ c, and hence may consistently be 2 c . Dobrinen et al [6] have also consistently constructed other 2 c -sized families of