Henry Sharp

Cardinality of finite topologies

Abstract Let S denote a finite set of cardinal n. The discrete topology on S contains 2n open sets; the indiscrete topology contains 2 open sets. A partial answer is given to the question: For which intermediate integers m is there a topology on S with cardinal m? It is shown that no topology, other than the discrete, has cardinal greater than 3/4 2n. Other bounds are derived on the cardinality of connected, non-T0, connected and non-T0, and non-connected topologies. Proofs involve results in the theory of transitive digraphs.

Score vectors of tournaments

Abstract A tournament T on any set X is a dyadic relation such that for any x , y ∈ X (a) ( x , x ) ∉ T and (b) if x ≠ y then ( x , y ) ∈ T iff ( y , x ) ∉ T . The score vector of T is the cardinal valued function defined by R ( x ) = |{ y ∈ X : ( x , y ) ∈ T }|. We present theorems for infinite tournaments analogous to Landaus necessary and sufficient conditions that a vector be the score vector for some finite tournament. Included also is a new proof of Landaus theorem based on a simple application of the “marriage” theorem.

Enumeration of vacuously transitive relations

The concept of vacuously transitive relation is defined and under an appropriate isomorphism, the equivalence classes of such relations are enumerated by use of the power group enumeration theorem [3]. This enumeration is shown to be combinatorially equivalent to a counting series derived by Harary and Prins [4] for certain kinds of bicolored graphs. Finally, it is shown that the main result can be extended to cover two additional cases of interest.

On families of finite sets with bounds on unions and intersections

We present a conjecture, with some supporting results, concerning the maximum size of a family of subsets satisfying the following conditions: the intersection of any two members of the family has cardinality at least s, and the intersection of the complements of any two members has cardinality at least r.