TL;DR
This paper provides estimates on the number of combinatorial designs, confirming Wilson's 1974 conjecture on Steiner Triple Systems, and introduces a simplified proof for clique decompositions in quasirandom graphs using Randomised Algebraic Construction.
Contribution
It generalizes Wilson's conjecture and offers a new, simpler proof for triangle decompositions in quasirandom graphs using a novel method.
Findings
Confirmed Wilson's conjecture on the number of Steiner Triple Systems.
Provided estimates on the count of combinatorial designs.
Simplified the proof of triangle decompositions in quasirandom graphs.
Abstract
We give estimates on the number of combinatorial designs, which prove (and generalise) a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. This paper also serves as an expository treatment of our recently developed method of Randomised Algebraic Construction: we give a simpler proof of a special case of our result on clique decompositions of hypergraphs, namely triangle decompositions of quasirandom graphs.
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Taxonomy
TopicsLimits and Structures in Graph Theory · graph theory and CDMA systems · Quasicrystal Structures and Properties