Design theory and finite combinatorics

Large Steiner Systems Construction Problem

Collaboration beta

The requested S(6,7,19) cannot exist. Any such design would yield an S(4,5,17), and the latter was ruled out by a peer-reviewed exhaustive classification-based search in 2008.

\exists S (6,7,19) ?

Known results and sources

Symbolic 19-element set and seven-block leading by fixed-pair derivation to the known nonexistence of S(4,5,17) and hence S(6,7,19).

Research problem

Exact mathematical statement

A Steiner system $S(6,7,19)$ would be a collection $\mathcal B\subseteq\binom{X}{7}$ on a 19-element set $X$ such that every 6-subset of $X$ lies in exactly one block. The exact question is whether such a system exists.

\exists\,S(6,7,19)\;?

The answer is no: fixing any pair in a hypothetical system and deleting that pair from every containing block would give an $S(4,5,17)$ , while Östergård and Pottonen proved that no $S(4,5,17)$ exists.

Problem infographic

Problem at a glance

Three-panel derivation from a hypothetical S(6,7,19) to an S(4,5,17), followed by the published nonexistence contradiction and exact block counts.

Current mathematical picture

Where work on Large Steiner Systems Construction Problem stands

Solved

The exact S(6,7,19) existence question is solved negatively. Östergård and Pottonen proved that no S(4,5,17) exists and state that their result rules out S(t,t+1,t+13) for every t≥4; taking t=6 gives nonexistence of S(6,7,19).

Reader boundaryHistorical research record

The retained argument stages below document an earlier investigation. They do not reopen this solved question or advertise new contribution tasks.

Archived quantitative snapshot

The retained investigation in numbers

This reconstructs the source's normalized mathematical content at the archived current snapshot. It describes historical material, not current work or progress toward an already solved question.

5,190current normalized lines of mathematical investigationArchived source snapshot

Argument development: 4,363 · 84%
Explored or eliminated routes: 108 · 2%
Computational analysis: 360 · 7%
Open obligations: 202 · 4%
Definitions and setup: 157 · 3%

How to interpret the archived roles

The role composition is an exact reconstruction from the retained content map. Labels such as open obligations describe the historical source classification; they are not current ProofAtlas tasks or invitations to continue this solved question.

Archived route inventory

Historical routes retained from the investigation

These records preserve how the argument was explored before the external solution. They are archival: none is a current ProofAtlas route or an invitation to continue the solved question.

1 retained record

Historical route recordIncomplete local construction data

The source requires all 476 or all 3,876 blocks plus an implementation-independent verifier for a positive result. The historical finite-search data remains useful for provenance, but it is unnecessary for the settled negative conclusion.

Archived disposition · Narrowed route

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026

Current statusSolved

[1][2]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
Peer reviewedA recent peer-reviewed design-theory article continues to cite the nonexistence of S(4,5,17) as an established input.[2]
2008
Peer reviewedÖstergård and Pottonen proved that no S(4,5,17) exists and explicitly noted that this rules out S(t,t+1,t+13) for every t≥4, including S(6,7,19).[1]

Mathematical neighborhood

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