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Topics in Linear Spaces and Projective Planes


Fowler, Joel Christopher (1984) Topics in Linear Spaces and Projective Planes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/d8gp-ka37.


A linear space is an incidence structure of points and lines such that every pair of points is contained in a unique line. In the first two chapters of this thesis results are presented linking structural properties to arithmetic conditions on the number of points and lines. We provide a short new proof of Jim Totten's classification of all linear spaces for which the difference between the number of points and lines does not exceed the square root of the number of points. We extend this classification when the number of points is of a certain form. Also in these chapters we have similar classification results for more specialized finite geometrical structures such as (r,l)-designs.

The last chapter is devoted to (k,u)-arcs. A (k,u)-arc in a finite projective plane is a set of k points meeting no line of the plane in more than u points. Elementary bounds upon k can be established and we call an arc with this maximum number of points perfect. An arc not properly contained in any other is called complete. Several constructions are given for both perfect and complete arcs. The major results of this chapter concern the uniqueness of completions of a (k,u)-arc to a perfect arc.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Wilson, Richard M.
Thesis Committee:
  • Wilson, Richard M. (chair)
  • Ryser, Herbert J.
  • Lorden, Gary A.
  • Dean, Richard A.
  • Wales, David B.
Defense Date:9 April 1984
Record Number:CaltechTHESIS:10302018-173228813
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:11254
Deposited By: Mel Ray
Deposited On:31 Oct 2018 17:53
Last Modified:16 Apr 2021 23:14

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