Steiner Systems — open problem in combinatorics

Steiner Systems

Construct an S(t, k, n)-Steiner system with n > k > t > 5, t < 10, and n < 200. No example of a Steiner system with t > 5 is known, despite a 2014 existence theorem by Keevash showing that such systems must exist for sufficiently large n. Reference: Large Steiner Systems

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

Start working on it Submit a claim Follow

Cite

@misc{cairn-steiner-system,
  title        = {Steiner Systems},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/steiner-system}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Construct an $S (t, k, n)$ -Steiner system with $n > k > t > 5$ , $t < 10$ , and $n < 200$ .

No example of a Steiner system with $t > 5$ is known, despite a 2014 existence theorem by Keevash showing that such systems must exist for sufficiently large $n$ .

Reference: Large Steiner Systems

A Steiner system $S (t, k, n)$ is a collection of $k$ -element subsets (called blocks) of an $n$ -element set such that every $t$ -element subset is contained in exactly one block.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.SteinerSystem.

theorem large_steiner_systems : Nonempty LargeSteinerSystemWitness

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
Large Steiner Systems by Kunal Marwaha

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.