Semiregular Large Sets

learned many things in the last several years that have helped me in several areas of this thesis. Along with the faculty, I would like to thank the office staff who do much in the department that is unseen, and have certainly done much for me personally. I would also like to thank Phil Romig for teaching me too many things to mention. Also, I would like to thank the members of my committee, Professors Spyros Magliveras, Douglas Stinson, and Jean-Camille Birget. They have taken time out of their busy schedules to read and comment on drafts of the thesis. I would like to thank my adviser, Professor Spyros Magliveras. He has taught me many things in the last 3 years, and has sparked my interest in several areas of research. He encourages me both personally and professionally. Lastly, I would like to thank my Lord and Savior, Jesus Christ, who has been the constant in my life as a student, and will continue to be when I am no longer a student. He has revealed to me a another part of His Big Book of Theorems. He gives meaning and purpose to my life and work, and has saved me from that which I could not save myself. To God be the glory! Adviser: Spyros Magliveras A t − (v, k, λ) design (X, B) is a v-element set X of points and a collection B of k-element subsets of X called blocks such that every t-element subset of X is contained in precisely λ blocks. The collection X k of all k-subsets of a set X forms a t-(v,k,λ) design with λ=λ= v−t k−t , and is called the complete design. N i=1 , of the complete design into N disjoint t − (v, k, λ/N) designs. These are also called large sets of t − (v, k, λ) designs. A group G is said to be an automorphism group of a large set B if B g =B for all g ∈ G. That is, if B g i ∈ B for all B i ∈ B and g ∈ G. Equivalently, we say that a large set with this property is G-invariant. If the stronger condition that B g i =B i for all B i ∈ B and g ∈ G holds, we say that a large set is [G]-invariant. If the G-orbits of a …