Matroid Correlation
Constants

← all families

certified lower bound · dashed curve

Steiner-system sparse-paving extensions

Sparse-paving matroids from Steiner systems give nonuniform 2-design bases and distinguished-pair values approaching \(\frac43\).

parameter

Steiner systems \(S(r-1,r,n)\)

supremum

supremum 4/3 — an iterated limit

discovered by

Chris Eur

discovery date

August 2026

Construction

Take the blocks of a Steiner system \(S(r-1,r,n)\) as the circuit-hyperplanes of a rank-\(r\) sparse-paving matroid \(M_{r,n}\), then form \(P_{r,n}=(M_{r,n}\oplus M_{r,n})+e\times f\). The plotted series use Steiner triple and quadruple systems, whose admissible orders are explicit.

What the curve shows

For fixed \(r\), the formula tends to \(4r/(3r+1)\) as \(n\to\infty\); those limits tend to \(\frac43\) as \(r\to\infty\). Keevash's design-existence theorem supplies the general fixed-rank families at all sufficiently large admissible orders.

Proof status

The adjacent-rank counts prove the displayed pair value exactly; full maximization remains open. Full proof on the mathematics page →

AI disclosure

Chris Eur used ChatGPT 5.6 Sol Max on Codex as a mathematical exploration and formula-checking aid while identifying the family, beginning with Steiner triples. A public chat link is not currently recorded.

Plotted members

S(2,3,n) (23 plotted members)

S(3,4,n) (23 plotted members)