Go to  Advanced Search

Please note that cIRcle is currently being upgraded to DSpace v5.1. The upgrade means that the cIRcle service will not be accepting new material from 05:00 on September 1/15 until 08:00 on September 4/15. Read only access will be available during this period. Apologies for the inconvenience.

Families of forbidden configurations

Show full item record

Files in this item

Files Size Format Description   View
ubc_2013_spring_koch_christina.pdf 499.8Kb Adobe Portable Document Format   View/Open
Title: Families of forbidden configurations
Author: Koch, Christina Louise
Degree Master of Science - MSc
Program Mathematics
Copyright Date: 2013
Publicly Available in cIRcle 2013-04-17
Abstract: The forbidden configuration problem arises from a question in extremal set theory. The question seeks a bound on the maximum number of subsets of an m-set given that some trace is forbidden. In terms of hypergraphs, we seek the maximum number of edges on a simple hypergraph of m vertices such that this hypergraph does not contain a forbidden sub-hypergraph. We will use the notation of matrices to describe the problem as follows. We call a (0,1)-matrix simple if it has no repeated columns; this will be the analogue of a simple hypergraph. Let F be a given matrix. We say that a (0,1)-matrix A contains F as a configuration if there is some submatrix of A that is a row and column permutation of F. This is equivalent to a hypergraph containing some sub-hypergraph. F need not be simple. We define forb(m, F ) as the maximum number of columns possible for a simple, m-rowed, (0,1)-matrix that does not contain F as a configuration. A variation of the forbidden configuration problem forbids a family of configurations F = {F₁, F₂, ... Fn} instead of a single configuration F. Thus, forb(m, F) becomes the maximum number of columns possible for a simple, m-rowed, (0,1)-matrix that does not contain any one of the matrices in the family as a configuration. We will present a series of results organized by the character of forb(m, F). These include a classification of families such that forb(m, F) is a constant and the unexpected result that for a certain family, forb(m, F) is on the order of m³/², where previous results typically had forb(m, F) on the order of integer powers of m. We will conclude with a case study of families of forbidden configurations and suggestions for future work.
URI: http://hdl.handle.net/2429/44225
Scholarly Level: Graduate

This item appears in the following Collection(s)

Show full item record

All items in cIRcle are protected by copyright, with all rights reserved.

UBC Library
1961 East Mall
Vancouver, B.C.
Canada V6T 1Z1
Tel: 604-822-6375
Fax: 604-822-3893