#### Title

Outline and nearly outline triple systems of even index

#### Semester

Fall

#### Date of Graduation

1998

#### Document Type

Dissertation

#### Degree Type

PhD

#### College

Eberly College of Arts and Sciences

#### Department

Mathematics

#### Committee Chair

Anthony J. W. Hilton.

#### Abstract

The main result of this work describes a completely novel structural result for (v, 3, lambda)-designs in the case when lambda is even. Such designs are known as triple systems. Given a finite collection of n elements, a triple system of order n and index lambda is a collection of subsets of size three chosen from the given set with the property that each distinct pair of elements from the set occurs in exactly lambda of the triples. It is well known for what values of n and lambda such systems exist. A partial triple system of order n and index lambda is a collection of triples from a set of size n with the property that each distinct pair of elements occurs in at most lambda triples. In 1975, C. C. Linder conjectured that every partial triple system of order n and index lambda = 1 can be embedded in a complete triple system of order m and index lambda = 1 where m is admissible and m ≥ 2n + 1. This conjecture extends readily to all integers lambda > 0. A recent result of Johansson showed that this conjecture is true when lambda is even. This work generalizes Johansson's result using the technique of amalgamations.;A triple system of order n and index lambda is thought of as a decomposition of the lambda-fold complete graph on n vertices into edge disjoint triangles. We amalgamate a set of vertices of a graph by identifying the vertices at a single vertex and preserving all edge adjacencies at the new vertex. Edges between amalgamated vertices become loops on the new vertex. The approach normally taken when using the process of amalgamations is to identify the characteristics possessed by an amalgamated structure and then define a new structure, referred to as an outline structure, that has those exact properties. The problem of an embedding is then reduced to determining when every such outline structure is in fact the amalgamation of a structure of the original type. The main result of this work is that every outline triple system of even index is the amalgamation of some complete triple system of the same index.

#### Recommended Citation

Ferencak, Michael Neill, "Outline and nearly outline triple systems of even index" (1998). *Graduate Theses, Dissertations, and Problem Reports*. 3146.

https://researchrepository.wvu.edu/etd/3146