Steiner triple systems satisfying the 4-vertex condition

Authors:
Petteri Kaski;Mahdad Khatirinejad;Patric R. Östergård
Affiliations:
Department of Information and Computer Science, Helsinki Institute of Information Technology HIIT, Aalto University, Aalto, Finland 00076;Department of Communications and Networking, Aalto University, Aalto, Finland 00076 and Department of Mathematics, University of British Columbia, Vancouver, Canada V6T 1Z2;Department of Communications and Networking, Aalto University, Aalto, Finland 00076
Venue:
Designs, Codes and Cryptography
Year:
2012

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Abstract

Higman asked which block graphs of Steiner triple systems of order v satisfy the 4-vertex condition and left the cases v = 9, 13, 25 unsettled.We give a complete answer to this question by showing that the affine plane of order 3 and the binary projective spaces are the only such systems. The major part of the proof is to show that no block graph of a Steiner triple system of order 25 satisfies the 4-vertex condition.