An account of some aspects of combinatorial mathematics by L. Mirsky

By L. Mirsky

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Pym (2). t Figures in bold-face type refer to the bibliography at the end of the book. 2 Hall’s Theorem and the Notion of Duality In the present chapter we initiate the study of combinatorial problems by proving P. Hall’s classical theorem on ‘distinct representatives’. As the discussion proceeds, we shall recognize that the whole of transversal theory may be regarded as a natural development of Hall’s investigation. 1 Transversals, representatives, and representing sets Let 2l = (Ai: i E I) be a family of subsets of a ground set E.

Hence M is maximal in 8. Using the alternative form of Zorn’s lemma relating t o minimal elements, we obtain in the same way the following result. L e t 5 be a (non-empty) collection of sets and suppose that, for each chain W in 8, n ( X : X E%) E 5, Then, for each A E 3, there exists u minimal element N E 8 such that N G A. Next, we introduce a concept which will play a fundamental role in much of the subsequent discussion. Suppose that a collection G of sets is such that a set belongs to E if and only if all its finite subsets belong t o G.

Show that Y* is maximal if and only if State the associated results for families of sets. 38 HALL’S THEOREM A N D THE NOTION OF DUALITY Notes on Chapter 2 S; 2. I . Transversal theory has not yet acquired a standardized terminology, and expressions such as ‘system of distinct representatives’, ‘common transversal’, ‘common system of representatives’ and so on are used in varying senses by different authors. Our own choice has primarily been guided by the need to avoid confusion between sets and families.

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