A very general theorem on systems of distinct representatives

Abstract

1. Introduction. Currently the theory of systems of distinct representatives (and the closely allied theory of transversals) is being carefully examined and reworked, often in a more general context which allows for the transfinite situation. This theory can be said to have had its beginning in 1935 when P. Hall proved his now celebrated theorem for the existence of a system of distinct representatives of a finite family of sets. In a no less significant paper M. Hall, Jr. (in 1948) extended P. Hall's theorem to infinite families of finite sets. Around these two theorems a considerable literature has grown (for an excellent survey and thorough bibliography see [10]). The two theorems have been refined in various ways by requiring that the system of distinct representatives have additional properties. It is however true that these refinements can be obtained by applying the original theorems to a modified family of sets. For finite families this is implicit in the work of Ford and Fulkerson [3] who show how most of these refinements can be obtained from their maximum flow-minimum cut theorem for flows in networks. For finite or infinite families Mirsky and Perfect [10], [11] have shown how these refinements can be obtained from the original theorems of the two Halls and a generalization of a mapping theorem of Banach [1]. In a recent paper [2] we obtained a further generalization of Banach's mapping theorem. This theorem along with M. Hall's theorem enables us to prove a very general theorem on systems of distinct representatives, which is in fact a transfinite and symmetrized form of a theorem of A. J. Hoffman and H. W. Kuhn. The theorem we prove contains as special cases (that is, without further refinement) all theorems that we know which assert the existence of a system of distinct representatives of a given family of sets or subfamily thereof with certain properties being required. We then can prove a theorem giving necessary and sufficient conditions that a family of sets possess a family of subsets whose cardinalities lie within prescribed bounds and where the frequencies of occurrences in these subsets of the elements lie within prescribed bounds. This will be made more precise later. From this we also obtain an extension to locally finite of Ore's solution [12] of the so-called subgraph problem for directed graphs and for that matter a generalization of Ore's solution due to Ford and