The authors enumerate the non-isomorphic Steiner triple systems of orders 19, 27 and 31 that contain Veblen points: 3, 1736, and 2 with exactly three Veblen points, plus partial counts for order 31 with one Veblen point.
Steiner Triple Systems of Order 21 with Subsystems
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abstract
The smallest open case for classifying Steiner triple systems is order 21. A Steiner triple system of order 21, an STS(21), can have subsystems of orders 7 and 9, and it is known that there are 12,661,527,336 isomorphism classes of STS(21)s with sub-STS(9)s. Here, the classification of STS(21)s with subsystems is completed by settling the case of STS(21)s with sub-STS(7)s. There are 116,635,963,205,551 isomorphism classes of such systems. An estimation of the number of isomorphism classes of STS(21)s is given.
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2024 1verdicts
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On the number of small Steiner triple systems with Veblen points
The authors enumerate the non-isomorphic Steiner triple systems of orders 19, 27 and 31 that contain Veblen points: 3, 1736, and 2 with exactly three Veblen points, plus partial counts for order 31 with one Veblen point.