Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Resolving Triple Systems into Regular Configurations. | Resolving Triple Systems into Regular Configurations E. Mendelsohn Department of Mathematics University of Toronto Toronto ON M5S 3G3 CANADA mendelso@ G. Quattrocchi Dipartimento di Matematica Universita di Catania Catania ITALIA quattrocchi@ Submitted June 25 1999 Accepted November 22 1999 Abstract A A Triple System v or a A-TS V B is a pair V B where V is a set and B is a subset of the 3-subsets of V so that every pair is in exactly A elements of B. A regular configuration on p points with regularity p on l blocks is a pair P L where L is a collection of 3-subsets of a usually small set P so that every p in P is in exactly p elements of L and L l. The Pasch configuration 0 1 2 3 4 5 012 g3s 245 134 has p 6 l 4 and p 2. A A-TS V B is resolvable into a regular configuration C P L or C-resolvable if B can be partitioned into sets lb so that for each i V Hj is isomorphic to a set of vertex disjoint copies of P L . If the configuration is a single block on three points this corresponds to ordinary resolvability of a Triple System. In this paper we examine all regular configurations C on 6 or fewer blocks and construct C-resolvable A-Triple Systems of order v for many values of v and A. These conditions are also sufficient for each C having 4 blocks or fewer. For example for the Pasch configuration A 0 mod 4 and v 0 mod 6 are necessary and sufficient. MRSC 05B07 1 Introduction The study of the way in which small configurations are germane to analysing the structure of combinatorial objects has progressed from the study of finite geometries 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R2 2 7 for example Desargues and Pappus configurations to using small configurations in the analysis of other designs. The concepts of avoidance of 1 13 ubiquity of 16 decomposability into 10 and bases for 9 small configurations have all provided insights into the structure of designs. On the other hand resolvability and A-resolvability have had a .
