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Steiner trigraphical designs and a block -size bound
A primary problem in combinatorial design theory is to determine when designs exist with prescribed properties. Here we focus on proper Steiner t-wise balanced designs. A proper Steiner t-wise balanced design (tBD) of type t-(v, K , 1) is a pair ( X , B ) where X is a v-element set of points, K is a subset of integers strictly between t and v, and B is a collection of subsets of X , called blocks, with the property that the size of every block is in K and every t-element subset of X is contained in exactly one block. For t ≥ 2, it is conjectured that if t is even then the maximum block size in a proper Steiner tBD is (v-1)/2 and if t is odd the maximum block size is v/2. This conjecture has previously been proven for t = 2, 3, 4 and 5. We prove that a proper Steiner tBD for t = 6 has a maximum block-size of v/2. We then attempt to determine all proper Steiner trigraphical tBD's. Here the v points of our design are the n3 triangles contained in the complete tri-partite graph Kn,n,n and the automorphism group of Kn,n,n, and of our design, is the wreath product Sn wr S3. There are exactly seven designs in the range 1 ≤ t ≤ 3 and no designs in the range 4 ≤ t ≤ 16. We conjecture that there are no proper Steiner trigraphical tBD's for t ≥ 4. We establish constraints on the size of n given t. ^
Ira, Michael S, "Steiner trigraphical designs and a block -size bound" (2000). ETD collection for University of Nebraska - Lincoln. AAI9967378.