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Combinatorics 1984: Finite Geometries and Combinatorial by A. Barlotti, etc., M. Biliotti, G. Korchmaros, G. Tallini

By A. Barlotti, etc., M. Biliotti, G. Korchmaros, G. Tallini

Curiosity in combinatorial innovations has been significantly stronger by way of the functions they might supply in reference to machine expertise. The 38 papers during this quantity survey the state-of-the-art and record on contemporary ends up in Combinatorial Geometries and their applications.Contributors: V. Abatangelo, L. Beneteau, W. Benz, A. Beutelspacher, A. Bichara, M. Biliotti, P. Biondi, F. Bonetti, R. Capodaglio di Cocco, P.V. Ceccherini, L. Cerlienco, N. Civolani, M. de Soete, M. Deza, F. Eugeni, G. Faina, P. Filip, S. Fiorini, J.C. Fisher, M. Gionfriddo, W. Heise, A. Herzer, M. Hille, J.W.P. Hirschfield, T. Ihringer, G. Korchmaros, F. Kramer, H. Kramer, P. Lancellotti, B. Larato, D. Lenzi, A. Lizzio, G. Lo Faro, N.A. Malara, M.C. Marino, N. Melone, G. Menichetti, okay. Metsch, S. Milici, G. Nicoletti, C. Pellegrino, G. Pica, F. Piras, T. Pisanski, G.-C. Rota, A. Sappa, D. Senato, G. Tallini, J.A. Thas, N. Venanzangeli, A.M. Venezia, A.C.S. Ventre, H. Wefelscheid, B.J. Wilson, N. Zagaglia Salvi, H. Zeitler.

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Every M . induces a partition of the point set of S ) . The proof of this theorem will be prepared by several Lemmas. 1. By a c l a w we mean a set S of lines with the following properties: (a) H { s f and H is parallel to every line of 5, (b) any two lines of S intersect. A claw S is called n o r m a l , if ( s l = d. The o r d e r of a claw is the number of its elements. A c l i q u e is a set of mutually parallel lines of S. The clique h is called m a x i m a l , if (a) H € M , ( b ) h is a maximal set of mutually parallel lines, and (c) Ihl > n - (d-l)(c+l) + x.

N . ,,rd s h a r p l y t - t r a n s i of H and t i v e on H: Consider t d i s t i n c t elements y11, moreover t d i s t i n c t e l e m e n t s y l j o , . . , y t j o of H . ,t , be t d i s t i n c t p o i n t s such t h a t no two o f them a r e c o m p e t i t o r s and such t h a t x Pl - Y P 1 ' for = Y pjo .. ,t . as w a s d e s c r i b e d a t t h e b e g i n n i n g B. Let u s have m a t r i c e s M 1 , . . , M n of A . Suppose n > 1. Then a l s o Mi,.. ,Mn-1 s a t i s f y ( i i ) and a s t r u c t u r e T ( r , q , r , n ) thus leads t o a s t r u c t u r e T ( r , q , r , n - l ) .

North-Holland) 39 EMBEDDING FINITE LINEAR SPACES IN PROJECTIVE PLANES Albrecht Beutelspacher and Klaus Metsch Fachbereich Mathematik der Universitat Saarstr. 21 D-6500 Mainz Federal Republic of Germany It is shown that a finite linear space in which all points have degree n+l can be embedded in a projective plane of order n, provided that the line sizes are big enough. INTRODUCTION A l i n e a r s p a c e is an incidence structure S = (p,L,I) of points, lines and incidences such that any two distinct points are on a unique line and any line contains at least two points.

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