Matroid Correlation
Constants

← all families

certified lower bound · dashed curve

Uniform self-sum extensions

A finite matroid for every integer \(r\ge2\), with a certified distinguished-pair value tending to \(\frac{4}{3}\) from below.

parameter

integer \(r\ge 2\)

supremum

supremum 4/3 — approached, never attained

discovered by

Alexander Divoux, Shouda Wang

discovery date

August 2026

Construction

Start with two copies of \(U_{r,2r}\), take their direct sum, freely extend by \(e\), and then freely coextend by \(f\). The resulting matroid \(P_r\) has \(4r+2\) elements and rank \(2r+1\).

What the curve shows

Exact basis counts for the distinguished pair \((e,f)\) give \(\frac{4r^2}{3r^2+2r+1}\), so every plotted point is a certified lower bound, and the bounds approach \(\frac{4}{3}\) as \(r\to\infty\); no finite \(r\) attains it. Computational remark (this site): exact enumeration confirms \((e,f)\) is the unique maximizing pair for \(2\le r\le 5\), so the curve is exact on those members — \(P_5\) is the record for 22 elements.

Proof status

The adjacent-rank identity proves the displayed pair value and its limit; maximization over all pairs remains open for general \(r\). Full proof on the mathematics page →

AI disclosure

Alexander Divoux and Shouda Wang used ChatGPT 5.6 Sol Ultra to assist the discovery of the original transversal-matroid family. Its reinterpretation as a double free-extension is due to humans. No public chat link is currently recorded.

Plotted members

Pᵣ (29 plotted members)