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Báo cáo toán học: "Steiner Triple Systems Intersecting in Pairwise Disjoint Blocks"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài:Steiner Triple Systems Intersecting in Pairwise Disjoint Blocks. | Steiner Triple Systems Intersecting in Pairwise Disjoint Blocks Yeow Meng Chee Netorics Pte Ltd 130 Joo Seng Road 05-02 Olivine Building Singapore 368357 ymchee@alumni.uwaterloo.ca Submitted Feb 18 2003 Accepted Mar 12 2004 Published Apr 2 2004 Abstract Two Steiner triple systems X A and X B are said to intersect in m pairwise disjoint blocks if A n B m and all blocks in A n B are pairwise disjoint. For each v we completely determine the possible values of m such that there exist two Steiner triple systems of order v intersecting in m pairwise disjoint blocks. 1 Introduction A set system is a pair X A where X is a finite set of points and A is a set of subsets of X called blocks. Let K be a set of positive integers. The set K is a set of block sizes for X A if A E K for every A E A. Let X A be a set system and let Q G1 . Gs be a partition of X into subsets called groups. The triple X Q A is a group divisible design GDD when every 2-subset of X not contained in a group appears in exactly one block and A n G 1 for all A E A and G E Q. We denote a GDD X Q A by K-GDD if K is a set of block sizes for X A . The group type or simply type of a GDD X Q A is the multiset G G E Q . When more convenient we use the exponential notation to describe the type of a GDD A GDD of type gl1 gS is a GDD where there are exactly tị groups of size g . If Q is not specified in a GDD X Q A it is taken that all groups are of size one Q x x E X . A 3 -GDD of type F is a Steiner triple system of order v and is denoted by STS v . Two GDDs D1 X Q1 A1 and D1 X Q2 A2 of the same type are said to intersect in m blocks if A1 n A2 m. If in addition the blocks in A1 n A2 are pairwise disjoint that is for any A A E A1 n A2 A A we have A n A 0 then D1 and D2 THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R27 1 are said to intersect in m pairwise disjoint blocks. Define Int T m I 3 two 3 -GDDs of type T intersecting in m blocks and Intd T m I 3 two 3 -GDDs of type T intersecting in m pairwise disjoint .