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# 18: More Designs

• • Joy Morris
• Professor (Mathematics) at University of Lethbridge
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• 18.1: Steiner and Kirkman Triple Systems
In 1847, Rev. Thomas Kirkman (previously mentioned in Chapter 13) found a complete solution to this problem in the case where the design is balanced (with λ=1), and made some progress towards solving the complete problem. Although Steiner did not study triple systems in 1853, he came up with Kirkman’s result independently, and his work was more broadly disseminated in mathematical circles, so these structures still carry his name.
• 18.2: t-Designs
In a BIBD, every pair appears together λ times. In the notation of Woolhouse’s problem, q=2. What about larger values of q? (We’ll still only consider the case where every q-set appears an equal number of times λ, so the design must be balanced, but we will include the more general situation that λ≥1.)
• 18.3: Affine Planes
You are probably familiar with at least some of Euclid’s axioms of geometry. If you haven’t taken geometry classes in university, you may not know that we can apply these axioms to finite sets of points, and discover structures that we call finite Euclidean geometries, or more commonly, affine planes. To avoid some trivial situations, we also require that the structure has at least three points, and that not all of the points lie on a single line.
• 18.4: Projective Planes
A projective plane is another geometric structure (closely related to affine planes). In a finite projective plane, the set of points (and therefore the set of lines) must be finite. Like finite affine planes, finite projective planes can be thought of as a special kind of design. As in the case of affine planes, the final axiom has been developed to avoid some trivial situations.
• 18.5: Summary
This page contains the summary of the topics covered in Chapter 18.