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# Steiner trigraphical designs and a block -size bound

#### Abstract

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* (*t*BD) 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 *t*BD 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 *t*BD for *t* = 6 has a maximum block-size of *v*/2. We then attempt to determine all proper Steiner trigraphical * t*BD's. Here the *v* points of our design are the * n*^{3} triangles contained in the complete tri-partite graph *K _{n,n,n}* and the automorphism group of

*K*, and of our design, is the wreath product

_{n,n,n}*S*wr

_{n}*S*

_{3}. 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

*t*BD's for

*t*≥ 4. We establish constraints on the size of

*n*given

*t*. ^

#### Subject Area

Mathematics

#### Recommended Citation

Ira, Michael S, "Steiner trigraphical designs and a block -size bound" (2000). *ETD collection for University of Nebraska - Lincoln*. AAI9967378.

http://digitalcommons.unl.edu/dissertations/AAI9967378