lines, and each line pencil has exactly elements. In particular, the number v is congruent to either 1 or 3 modulo 6. In fact, the following classical theorem holds.

**Theorem.** There exists a Steiner triple system on v points
if and only if
.

The number of isomorphism types of the small Steiner triple systems is well known.

v | isomorphism types |
---|---|

1 | 1 |

3 | 1 |

7 | 1 |

9 | 1 |

13 | 2 |

15 | 80 |

19 | >2000000 |

It's commonly assumed that the number of isomorphism types of Steiner
triple systems with v=19 points is about 10^{9}. But unfortunately,
this is out of the range of todays computers.

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