By L. Mirsky

ISBN-10: 0124985505

ISBN-13: 9780124985506

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**Additional resources for An account of some aspects of combinatorial mathematics**

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2 F o r 1 < k < n , 2 < i 1 < ... < i k < n , w e h a v e ] B i lu ... u Bi,l = / ( A i ,u ... LJ Aik)\ (x,}( 3 JAi,u ... u Aikl - I{xl}I - (Ail u ... u Ai,J - 1 3 ( k + 1) - 1 = k . e. , x, E B,. , x, are distinct. , x,}+ is a transversal of a. Case 2. Suppose that, for some k with 1 1 < i, < ... < ik < n, we have c A,,

Show that there exists a collection 8 of subsets of X and a bijection $: X + 3 such that, for any elements x, y E X, we have x < y if and only if $(x) c $( y ) . 7. Let X be a set and let < , < * be two partial orders defined on X. We say that < * is an ‘extension’ of < if x < y implies x < * y . Verify that the collection P of all partial orders on X extending a given partial order is partially ordered by extension. Show that every chain in P has an upper bound, and deduce that every partial order on X can be extended to a total order.

Menger (1). The most satisfactory proof of Menger’s theorem is probably that of J. S. 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.

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