---
tag: 7BPSC4T4M3
kind: concept
name: Steiner system
msc: [05B05, "51E10"]
specializes: [4JCSS2Z1BA]
sources:
  - url: https://en.wikipedia.org/wiki/Steiner_system
same_as:
  wikidata: Q4420916
  wikipedia: Steiner system
---

## Summary

A Steiner system $S(t, k, n)$ is a set of $n$ points with a family of $k$-element blocks.
Every $t$ points lie in exactly one block.
It is a [[block design|4JCSS2Z1BA]] with $\lambda = 1$.
Steiner triple systems, finite projective planes and the large Witt design $S(5, 8, 24)$ are the best-known examples.
The node [[Steiner system|QHPW53E7YA]] gives the precise definition.

## Explanations

### L1 human

A Steiner system is a way to group points into blocks of equal size so that any choice of $t$ points determines exactly one block.
With $t = 2$, two points always lie together in exactly one block, like two points on exactly one line.
The [[Fano plane|QQ8GW1ASZM]] is a Steiner system $S(2, 3, 7)$: $7$ points, $7$ blocks of size $3$, and every pair of points in one block.