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By L. Mirsky

ISBN-10: 0124985505

ISBN-13: 9780124985506

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Adjunction. The sets in the family 2l of subsets of E are enlarged by the adjunction of ‘dummy’ elements which do not belong to the ground set E. Extension. The family ‘91 is extended by the addition of further sets. (Cf. ) Replication. We obtain a new family by taking a suitable number of copies of each set in 2l. Proliferation. A certain set A in 2l is replaced by a new set A‘ defined by the formula A‘ = { ( x , i ) : X E A ,1 < i < k,). ). , k ) and in that case proliferation simply reduces to the formation of Cartesian products.

The proof of this result depends on the simple observation that the complement of an incidence-bound resp. incidence-free collection is incidence-free resp. incidence-bound. ,xm, =p (4) + q. ,A n + + (5) is an incidence-free collection, so that t 3 (rn - p ) + (n - q) = m n - t,. Again, let (4) be a maximal incidence-free collection, so that t = p q. Then (5) is an incidence-bound collection and so t, < rn + n - f. Hence t, + t = m n, as required. 6. , A,, is incidence-free. + This theorem asserts that t* 2 r if and only if t < m + n - r ; in other words, that t* t = m + n.

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 , ... u A i k } and hence l{,j: I < j < pi, Bj n F # 011 = \Ai, u ... u Aikl 3 k = IF]. , B,,,); and this means that YI possesses a transversal. Finally, we mention a slight generalization of the preceding theorem. 5. , A,) be a family ofsubsets of E, and let E* 5 E.

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

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