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 \(S(1, 2, v)\).