L. Mirsky's An account of some aspects of combinatorial mathematics PDF

By L. Mirsky

ISBN-10: 0124985505

ISBN-13: 9780124985506

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3, the relation of inequality between cardinals is a partial order: it is also a total order, but this fact is much harder to prove. More obvious instances of total order come readily to mind. Thus any set of real numbers is totally ordered with respect to the relation of (ordinary) inequality. If ( X , <) is a partially ordered set, then we can define another partial order, say < *, on X by declaring that x < * y if and only if y d x. We shall say that < * is the reciprocal order of < . Let (X, <) be a partially ordered set and let A be a subset of X.

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. If 1B1 3 2, then there exists an element x E B S M C ~that (Ai: i E I) + (B \ {x}) again satisfies Hall’s condition.

3. The theorem in question is as follows. 4. , A,,) be a family of subsets of E. e. I{i: 1 < i < n, A i n F # 0)l 2 IF1 foreach F 5 E. (3) The necessity of condition (3) is obvious and it only remains to establish its sufficiency. , x,}+. , n}. jl < ... < j , < m. Then B ~ u, ... u B . 2 PROOFS OF THE FUNDAMENTAL THEOREM FOR FINITE FAMILIES 31 and so, by (3), IBj, u ... u Bjkl 3 k . , n}. e. ,x,~)= E is a PT of (21. 4. , ik}+ c { 1, .. , n } , and the B’s are defined as above, then { j : 1 d j d m, B j n F # 0}= { j : 1 < j d r n , x j ~ A iu , ...

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

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