An explicit Steiner system with t ≥ 6

A Steiner system $S(t, k, n)$ is a collection of $k$-element blocks from $n$ points such that every $t$ points lie in exactly one block. Keevash proved in 2014 that such systems exist for every $t$ whenever $n$ is large and the obvious divisibility conditions hold, yet no example with $t \ge 6$ has ever been written down. The task is to find one of modest size.

Problem

An $S(t, k, n)$ Steiner system is a set $\mathcal{B}$ of $k$-element subsets ("blocks") of $\{1, \dots, n\}$ such that every $t$-element subset of $\{1, \dots, n\}$ is contained in exactly one block.

Task. Construct an explicit $S(t, k, n)$ with

$$n > k > t \ge 6, \qquad t \le 9, \qquad n < 200.$$

(Epoch writes these as $(n, q, r)$-systems with $q = k$ and $r = t$.)

Background

Questions about such designs go back to Plücker, Kirkman and Steiner in the nineteenth century, and Keevash's theorem answered Steiner's 1853 question about their existence [1]. His proof, and the later ones by Glock, Kühn, Lo and Osthus [2], by Delcourt and Postle [3] and by Keevash again [4], are probabilistic or absorption-based. They guarantee existence only for enormous $n$ and give no usable example. Explicit Steiner systems are known only for $t \le 5$, for instance the Witt designs $S(5, 6, 12)$ and $S(5, 8, 24)$. Explicit designs matter in practice, for experimental design and error-correcting codes.

What is known

• For all fixed $t < k$, $S(t, k, n)$ exists for all sufficiently large $n$ satisfying the divisibility conditions [1–4].
• No Steiner system with $t \ge 6$ has been exhibited (Epoch; also recorded in formal-conjectures).
• Epoch's problem author doubts that even an optimised brute-force search could succeed, and expects a new conceptual approach to be needed.

What would count as progress

• The parameters and the list of blocks. Checking is direct: either count, for each $t$-subset, the blocks containing it, or check that any two blocks share fewer than $t$ points and that there are exactly $\binom{n}{t} / \binom{k}{t}$ blocks.
• Epoch scores submissions with its own verifier, given $n$, $q = k$, $r = t$ and the blocks.
• formal-conjectures states exactly this task in Lean as large_steiner_systems in SteinerSystem.lean.
• A construction with a large symmetry group, which explains why it works, would likely point to further examples.

References

1. P. Keevash, The existence of designs, 2014. arXiv:1401.3665
2. S. Glock, D. Kühn, A. Lo and D. Osthus, The existence of designs via iterative absorption: hypergraph $F$-designs for arbitrary $F$, 2016. arXiv:1611.06827
3. M. Delcourt and L. Postle, Refined absorption: a new proof of the existence conjecture, 2024. arXiv:2402.17855
4. P. Keevash, A short proof of the existence of designs, 2024. arXiv:2411.18291

Source

Imported from FrontierMath: Open Problems, large-steiner-systems, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.