By A. Barlotti, M. Biliotti, A. Cossu, G. Korchmaros and G. Tallini (Eds.)
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Additional info for Combinatorics ’84, Proceedings of the International Conference on Finite Geometries and Combinatorial Structures
LINEAR SPACES WITH CONSTANT POINT DEGREE Let A be a finite set of nonnegative integers. We say that the linear space S is A-semiaffine, if r -k < A for every non-incident point-line pair (p,L) of S . TRe kinear space S is called A-affins, if it is A-semiaffine, but not A‘-semiaffine for every proper subset A ’ of A. Throughout this section, S will denote an A-affine linear space in which every point has degree n+l. Because the lO)-affine linear I 0 1 throughspaces are the projective planes, we will assume A out.
Since b = kL*n + I I I 1 , we have I I I l = n+l+z. Since t 5 n we have 5 v = 1 + t(c-1) + (n+l)(n-c) (n+l+z)(n+l-c= = I~~l(n+l-c) -< n2+l-c. Now we claim z 5 c-2. (Otherwise, we would have (n+c)(n+l-c) 5 n’+l-c,. so n 5 c2-2c+l = (c-l)2I contradicting ( * ) . I NOW (1) and (2) follow immediately from ( * ) . Hence S is embeddable in a projective plane P of order n. In particular, b 5 n2+n+l. Let L and n be as above. We distinguish two cases. II. C a s e 1 . All long lines are contained in Then t 5 1 and s o t = 1.
Journ. Geom. 8 (1976), 61-73. [ l o ] H. Schaeffer, Das von Staudtsche Theorem in der Geometrie der Algebren. J. reine angew. Math. 267 (1974), 133-142. V. (North-Holland) ON n-FOLD 31 B L O C K I N G SETS Albrecht Beutelspacher and Franco Eugeni Fachbereich Mathematik der Universitat Mainz F e d e r a l R e p u b l i c o f Germany I s t i t u t o Matematica Applicata Facolta' Ingegneria L'Aquila , I t a l i a An n - f o l d b l o c k i n g s e t i s a s e t o f n - d i s j o i n t b l o c k i n g s e t s .