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- W2734530020 abstract "This dissertation studies the ‘asymptoticexistence’ conjecture for 3-designs with the primary goal ofconstructing new families of 3-designs. More specifically, thisdissertation includes the following: Firstly, byconsidering the action of the group PSL(2,q) on the finiteprojective line and the orbits of the action of this group toconstruct simple 3-designs. While the case q congruent to 3 modulo4 is 3-homogeneous (so that orbits of any ‘base’ block’ would yielddesigns), the case q congruent to 1 modulo 4 does not work the sameway. We overcome some of these issues by considering appropriateunions of orbits to produce new infinite families of 3-designs withPSL(2,q) acting as a group of automorphisms. We also prove that ourconstructions actually produce an abundance of simple 3-designs forany block size if q is sufficiently large and also construct alarge set of Divisible designs as an application of ourconstructions. We generalize the notion of aCandelabra system to more general structures, called Rooted ForestSet systems and prove a few general results on combinatorialconstructions for these general set structures. Then, we specializeto the case of k=6 and extend a theorem of Hanani to produce newinfinite families of Steiner 3-designs with block size6. Finally, we consider Candelabra systems andprove that a related incidence matrix has full row rank over therationals. This leads to interesting possibilities for ‘lambdalarge’ theorems for Candelabra systems. While a ‘lambda large’theorem for Candelabra systems do not directly yield any Steiner3-design, it allows for constructions of new Steiner 3-designs onlarge sets using methods such as Blockspreading." @default.
- W2734530020 created "2017-07-21" @default.
- W2734530020 creator A5062000960 @default.
- W2734530020 date "2008-01-01" @default.
- W2734530020 modified "2023-09-27" @default.
- W2734530020 title "The 3-Design Problem" @default.
- W2734530020 hasPublicationYear "2008" @default.
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