By L. Mirsky

ISBN-10: 0124985505

ISBN-13: 9780124985506

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1 that t* 3 t,. But any incidence-bound collection must contain at least t* objects. Hence t* < t * , and the proof is complete. 5. W e have t, + t = m + n. 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.

We shall conclude the present section by deriving a result which stands in a ‘dual’ relation to Hall’s theorem. 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}.

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 , ... 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.

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

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