Let P be a finite, connected partially ordered set containing no crowns and let Q be a subset of P. Then the following conditions are equivalent: (1) Q is a retract of P; (2) Q is the set of fixed points of an order-preserving mapping of P to P; (3) Q is obtained from P by dismantling by irreducibles.

In this paper we prove two analogues of the following theorem:

If is a collection of distinct subsets of {1, …, m} such that , i ≠ j ⇒ Ai ∩ Aj ≠ ∅, Ai ∪ Aj ≠ {1, …, m} then .

An asymptotic expansion is obtained for this sequence, of interest in combinatorial analysis. Values are given for the constants appearing in the leading term and a numerical comparison made.

The letters a, b, n, m, t (perhaps with suffixes) always denote natural numbers. A, B, S denote finite sets of natural numbers. |A| stands for the cardinality of A.

For a given constant c > 1 we say that the set A has property α(c) if there are at most c|A| differences a − b ≥ 0 for a, b ∈ A.

A. J. W. Hilton [5] conjectured that if P, Q are collections of subsets of a finite set S, with |S| = n, and |P| > 2n−2, |Q| ≥ 2n−2, then for some A ∈ P, B ∈ Q we have A ⊆ B or B ⊆ A. We here show that this assertion, indeed a stronger one, can be deduced from a result of D. J. Kleitman. We then give another proof of a recent result also proved by Lovász and by Schönheim.

Let En+1, for some integer n ≥ 0, be the (n + 1)-dimensional Euclidean space, and denote by Sn the standard n–sphere in En+1, . It is convenient to introduce the (–1)-dimensional sphere , where denotes the empty set. By an i-dimensional subsphere T of Sn, i = 0 n, we understand the intersection of Sn with some (i+1)-dimensional subspace of En+1. The affine hull of T always contains, with this definition, the origin of En+1. is the unique (–1)-dimensional subsphere of Sn. By the spherical hull, sph X, of a set , we understand the intersection of all subspheres of Sn containing X. Further we set dim X: = dim sph X. The interior, the boundary and the complement of an arbitrary set , with respect to Sn, shall be denoted by int X, bd X and cpl X. Finally we define the relative interior rel int X to be the interior of with respect to the usual topology sphZ . For each (n–1)-dimensional subsphere of Sn defines two closed hemispheres of Sn, whose common boundary it is. The two hemispheres of the sphere Sº are denned to be the two one-pointed subsets of Sº. A subset is called a closed (spherical) polytope, if it is the intersection of finitely many closed hemispheres, and, if, in addition, it does not contain a subsphere of Sn. is called an i-dimensional, relatively open polytope, , or shortly an i-open polytope, if there exists a closed polytope such that dim P = i and Q = rel int P. is called a closed polyhedron, if it is a finite union of closed polytopes P1 …, Pr. The empty set is the only (–1)-dimensional closed polyhedron of Sn. We denote by the set of all closed polyhedra of Sn. is called an i-open polyhedron, for some , if there are finitely many i-open polytopes Q1 …, Qr in Sn such that , and dim . By we denote the set of all i-open polyhedra. Clearly for all , and each i-dimensional subsphere of Sn, , belongs to and to , For each i-dimensional subsphere T of Sn, set . A map is defined by , for all , and, for all .

In [1], A. H. Stone proved that for a cardinal number k ≥ 1 a set with a transitive relation can be partitioned into k cofinal subsets provided each element of the set has at least k successors. Using methods quite different from those of Stone, we show that for k ≥ ℵ0 the same condition on successors guarantees that a set on which there are defined not more than k transitive relations can be partitioned into k sets each of which is cofinal with respect to each of the relations. We also show that such a partition exists even if some of the relations are not transitive as long as the non-transitive relations have no more than k elements.