Read e-book online An account of some aspects of combinatorial mathematics PDF

By L. Mirsky

ISBN-10: 0124985505

ISBN-13: 9780124985506

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E. Szpilrajn (l)] 8. Let S be a set of n elements and denote by T the collection of the 2” - 1 nonempty subsets of S, partially ordered by inclusion. Show that, for any antichain A in T, Deduce that the maximum cardinal of antichains in T is equal to where [4n] denotes the largest integer not exceeding +n. [Lubel (1); cf. Sperner (2)] 9. Do a maximal chain and a maximal antichain in a partially ordered set necessarily have a non-empty intersection? 6 Tychonoff’s theorem Since combinatorial analysis is primarily concerned with objects that have little or no structure, it draws in the main upon results in the theory of sets.

X n E B n s An, where x k + , .. , x, are distinct. , x,,} is a transversal of 91. The proof just given is extremely transparent and can, as we shall see, be adapted to serve in other situations (cf. g. 2 and Ex. 3). 2) and has the advantage of even greater simplicity. To describe it, we shall need two preliminary results; and i n view of subsequent applications, we formulate these for arbitrary rather than for finite families. 2. Let the family (Ai: i E I) + (B)? of sets satisfy Hall’s condition.

Now it is clear that a partial transversal is related to sets and elements in a symmetrical way. We might therefore expect some theorems involving partial transversals to be symmetric with respect t o these two collections of objects. 6 below are results of this kind. 1. The lack of notational symmetry, on the other hand, can be corrected by a mere change of notation (and terminology). We recall, then, that we are dealing with two classes of objects: the sets Ai, i ~ of1 the family 41 = ( A i : ig I), and the elements of the ground set E.

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


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