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2008, Doctor of Philosophy, Ohio State University, Mathematics.

This dissertation studies the ‘asymptotic existence’ conjecture for 3-designs with the primary goal of constructing new families of 3-designs. More specifically, this dissertation includes the following:

Firstly, by considering the action of the group PSL(2,q) on the finite projective line and the orbits of the action of this group to construct simple 3-designs. While the case q congruent to 3 modulo 4 is 3-homogeneous (so that orbits of any ‘base’ block’ would yield designs), the case q congruent to 1 modulo 4 does not work the same way. We overcome some of these issues by considering appropriate unions of orbits to produce new infinite families of 3-designs with PSL(2,q) acting as a group of automorphisms. We also prove that our constructions actually produce an abundance of simple 3-designs for any block size if q is sufficiently large and also construct a large set of Divisible designs as an application of our constructions.

We generalize the notion of a Candelabra system to more general structures, called Rooted Forest Set systems and prove a few general results on combinatorial constructions for these general set structures. Then, we specialize to the case of k=6 and extend a theorem of Hanani to produce new infinite families of Steiner 3-designs with block size 6.

Finally, we consider Candelabra systems and prove that a related incidence matrix has full row rank over the rationals. This leads to interesting possibilities for ‘lambda large’ theorems for Candelabra systems. While a ‘lambda large’ theorem for Candelabra systems do not directly yield any Steiner 3-design, it allows for constructions of new Steiner 3-designs on large sets using methods such as Block spreading.

Neil Robertson, PhD (Advisor)

Akos Seress, PhD (Committee Member)

Stephen Milne, PhD (Committee Member)

Akos Seress, PhD (Committee Member)

Stephen Milne, PhD (Committee Member)

114 p.

Balachandran, N. (2008). The 3-Design Problem. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

Balachandran, Niranjan. "The 3-Design Problem." Electronic Thesis or Dissertation. Ohio State University, 2008. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

Balachandran, Niranjan "The 3-Design Problem." Electronic Thesis or Dissertation. Ohio State University, 2008. https://etd.ohiolink.edu/