Steiner Systems

formal-conjectures · wikipedia · ams-5

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.

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

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/Steiner_system
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://epoch.ai/frontiermath/open-problems/large-steiner-systems
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/SteinerSystem.lean
Local target: Corpus.WikipediaSteinerSystem.large_steiner_systems
Source SHA-256: f5e038c8fc82fd6edefffbbd8bfa30e81958bfe95732c7c01ae101822e8d15c1
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record