# Steiner Systems

A Steiner system \(S(t,k,v)\) is a collection of k-sized subsets (“blocks”) of the numbers 1 to v. These collections are special because every t-sized subset of the numbers 1 to v are in exactly one block.

For example, here is a \(S(2,3,7)\) system (also known as the Fano plane):

```
{1,2,3}
{1,4,5}
{1,6,7}
{2,4,6}
{2,5,7}
{3,4,7}
{3,5,6}
```

I was introduced to Steiner systems from this review article. There is also good information on Dan Gordon’s page.

Some cool facts about Steiner systems:

- If you have a Steiner system \(S(t,k,v)\), you can make a Steiner system \(S(t-1, k-1, v-1)\) by choosing all blocks with a certain number, and deleting that number.
- Steiner systems follow divisibility rules: \(S(t,k,v)\) only can exist if \({k-i \choose t-i}\) divides \({v-i \choose t-i}\) for all \(i \in \{0,\cdots,t-1\}\).

## Table of \(S(t, t+1, v)\)

- “Trivial”: There is always a “trivial” Steiner system \(S(t,k,k)\) with one block including all numbers from 1 to k.
- “Pairs”: If \(v\) is even, then pairing all numbers from 1 to v forms a Steiner system \(S(1, 2, v)\).

\(t=1\) | \(t=2\) | \(t=3\) | \(t=4\) | \(t=5\) | \(t=6\) | |
---|---|---|---|---|---|---|

\(v=1\) | - | - | - | - | - | - |

\(v=2\) | Trivial | - | - | - | - | - |

\(v=3\) | ✖ | Trivial | - | - | - | - |

\(v=4\) | Pairs | ✖ | Trivial | - | - | - |

\(v=5\) | ✖ | ✖ | Trivial | - | - | |

\(v=6\) | Pairs | ✖ | Trivial | - | ||

\(v=7\) | ✖ | Fano | Trivial | |||

\(v=8\) | Pairs | ✖ | ||||

\(v=9\) | ✖ | |||||

\(v=10\) | Pairs | ✖ | ||||

\(v=11\) | ✖ | ✖ | ||||

\(v=12\) | Pairs | ✖ | ||||

\(v=13\) | ✖ | |||||

\(v=14\) | Pairs | ✖ |