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By L. Mirsky

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2 < n + min {IA(I)l - [I[}. I Moreover, 41 has no P T of cardinal t* IA(1)l < 111 for some I. e. 1, + t* + 1 - n + IA(1)I - 111 r* 3 n + min {IA(I)I - III}. I The assertion is therefore proved. 1. , x,}+. ’, A, (where two A’s are regarded as formally distinct even when they are equal as sets) will simply be called objects. A collection S of objects is said to be incidence-bound if the relation x i E Aj implies that at least one of x i , Aj belongs to S. Again, S is said to be incidence-free if x i \$ A j whenever both x i and Aj belong to S.

A, x R). By Hall’s theorem, 4I* has a transversal if and only if, for each I 111 d 1u is1 ( A i x R ) ! = IA(I) x RI = rlA(1)l. I* has a transversal if and only if (2) is satisfied. I* possesses a transversal, say (xi,k,) E A, x R, ... (x,,,kn)E A, x R, 7 where no two of the pairs (xi, k i ) are identical. , n>. [(Ij) = ( A i :i E Ij) possesses a transversal, Thus 21 can be partitioned into r subfamilies each of which possesses a transversal. I can be partitioned into r subfamilies each of which possesses a transversal.

Ore (3), and F. Harary (1). Ore is also the author of a short elementary introduction (4) to this branch of mathematics. (6), and (7,233). 1, see (59, by K. 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’.