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A density lemma

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Suppose we possess a theorem T about deltoids. Let 91 = ( A i : ic I ) and 23 = ( B j : . j € J ) be two families of sets and associate with them the deltoid 22 = (I, A, J), where ( i , , j ) E A if and only if A ; n B j # 0. Then, applying theorem T to 9, we obtain a result concerning the families 91 and 23. This process will be called syn7nretric interpretation. In what follows, we do not propose to adopt a uniform pattern of presentation. In many cases, we shall continue to speak in terms of sets and elements; in other cases we shall state the argument in deltoid form and derive statements about families of sets as immediate corollaries; and frequently we shall leave the interpretation or dualization to the reader.

1 is due to Knaster and Tarski; see Knaster (1); cf. also D. Konig (3) for a number of related results. 1). 3, proved by Banach (1) in 1924, may be regarded as the genesis of all results in the present section. 6). A particularly simple proof of this latter result will be found in J. L. Kelley’s treatise (1, 28). R. A. Brualdi (3) obtained generalizations of almost all results in this section. 1. 4. Most mathematicians are undoubtedly familiar with boolean atoms, but it is not altogether easy to give precise references.

We shall not be able to say a great deal about representing sets : questions concerned with them seem to be exceptionally difficult. The major part of our discussion will be devoted to (partial or total) transversals and to systems of representatives, possibly subject to certain restrictions and associated with one or more than one family. Nearly all problems we study are qualitative in the sense that they are concerned with questions of existence. Quantitative problems arise quite naturally and are of equal interest, but they are generally found to be intractable and we shall refer to them only on isolated occasions.

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