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

Current state (tracked automatically)

Formal Conjectures marks the Lean statement large_steiner_systems as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `large_steiner_systems`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 4 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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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 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.