They've been framed! (Colouring Kirkman triple systems)
Speaker: Nicholas Cavenagh
Affiliation: University of Waikato, New Zealand
Abstract
A Steiner triple system (STS) is a partition of a point set into triples so that each pair occurs in just one triple. If the triples can furthermore be partitioned into resolution classes so that each point is in precisely one triple in each resolution class, this is a Kirkman Triple system (KTS). These properties are useful in experimental design when we want to test pairwise interactions between treatments in an efficient way.
Frames are constructions which, when combined with specific STSs and KTSs, can be used to make KTSs of larger orders. We review the standard frame methods from the literature, and apply these to colourings KTSs. Our main result is that for any v congruent to 3 (mod 6), there exists a KTS on v points and a colouring of the points by 3 colours such that no block is monochromatic.
(Joint work with: Andrea Burgess, Peter Danziger and David Pike, "Weak colourings of Kirkman Triple System", Designs Codes and Cryptography, 2025)
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