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Combinatorial Mathematics VII: Proceedings of the Seventh by R. G. Stanton, J. G. Kalbfleisch (auth.), Robert W. PDF

By R. G. Stanton, J. G. Kalbfleisch (auth.), Robert W. Robinson, George W. Southern, Walter D. Wallis (eds.)

ISBN-10: 354010254X

ISBN-13: 9783540102540

ISBN-10: 354038376X

ISBN-13: 9783540383765

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Read Online or Download Combinatorial Mathematics VII: Proceedings of the Seventh Australian Conference on Combinatorial Mathematics Held at the University of Newcastle, Australia, August 20 – 24, 1979 PDF

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Extra info for Combinatorial Mathematics VII: Proceedings of the Seventh Australian Conference on Combinatorial Mathematics Held at the University of Newcastle, Australia, August 20 – 24, 1979

Sample text

Clearly F 1 contains every t-set not involving ~, F 3 contains every t-set involving =. Since F 1 + F 3 may not be as economical as possible, we have Theorem 2. 3. N(t,k,v) ,< N(t,k,v-l)+N(t-l,k-l,v-l). KNOWN RESULTS An extensive survey of results appears in [3]; a very excellent and up-to- date survey with many important new results appears in [2]. Briefly, the following results are known (for complete references, see the extensive reference list in [2]). L+2 (I) N(2,3,v) = L(2,3,v), a result proved by Fort and Hedlund.

2) N(2,4,v) = L(2,4,v) for v ~ 7,9,10,19 (the result is L + 1 for 7,9,10 ; for 19). (3) N(2,5,v) has been studied by Gardner and others; (4) N(k-l,k,v) has been studied; some results are known. many results are known, but the only com- plete determination is for N(3,4,v), with v ~ 7, mod 12. given by the bound L(3,4,v). In those cases, N(3,4,v) is For v _-- 7 rood 12, the only known result is that N(3,4,7) = 12, whereas the bound is ii. (5) Numerous Steiner Systems S(t,k,v) are known; ting appears in [2].

V I f n >~ ~ - a n d m > ( t - 1 ) n , = Ik_t+I l . than Form a d e s i g n on n b l o c k s w h i c h c o n t a i n s varieties; " then B ( t , k , m , v ) g n. an m - s e t w h i c h i s n o t c o v e r e d by t h i s e a c h b l o c k i n a t most t - 1 v-m k-t+l each variety design, it so we would h a v e m < n ( t - 1 ) . at least once. would h a v e t o meet This is a contra- diction, Lemma 3 . 9 . Proof. B(2,k,m,v) If I ~ I ¢ m. 8 shows that B(2,k,msV) < m. it If can be altered to produce a design containing every variety.

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Combinatorial Mathematics VII: Proceedings of the Seventh Australian Conference on Combinatorial Mathematics Held at the University of Newcastle, Australia, August 20 – 24, 1979 by R. G. Stanton, J. G. Kalbfleisch (auth.), Robert W. Robinson, George W. Southern, Walter D. Wallis (eds.)


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