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Properties of Steiner triple systems of order 21

Grahame Erskine, Terry S. Griggs

Abstract

Properties of the 62,336,617 Steiner triple systems of order 21 with a non-trivial automorphism group are examined. In particular, there are 28 which have no parallel class, six that are 4-chromatic, five that are 3-balanced, 20 that avoid the mitre, 21 that avoid the crown, one that avoids the hexagon and two that avoid the prism. All systems contain the grid. None have a block intersection graph that is 3-existentially closed.

Properties of Steiner triple systems of order 21

Abstract

Properties of the 62,336,617 Steiner triple systems of order 21 with a non-trivial automorphism group are examined. In particular, there are 28 which have no parallel class, six that are 4-chromatic, five that are 3-balanced, 20 that avoid the mitre, 21 that avoid the crown, one that avoids the hexagon and two that avoid the prism. All systems contain the grid. None have a block intersection graph that is 3-existentially closed.
Paper Structure (8 sections, 1 theorem, 2 figures, 2 tables)

This paper contains 8 sections, 1 theorem, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Parallel classes
  3. Colourings
  4. Configurations
  5. Pasch trades
  6. Other properties
  7. Extremal systems
  8. Further questions

Key Result

Theorem 2.1

(Kokkala and Östergård) The Steiner triple systems of order 21 with a non-trivial automorphism group underlie 66,937 non-isomorphic Kirkman triple systems with a non-trivial automorphism group and a further 1,992 Kirkman triple systems having only the identity automorphism. None of these systems is

Figures (2)

  • Figure 1: The full $(\ell+2,\ell)$ configurations on at most six blocks
  • Figure 2: The even $(9,6)$-configurations

Theorems & Definitions (1)

  • Theorem 2.1