White’s symmetric-exchange conjecture for matroid bases

White’s conjecture asks whether the toric ideal of a matroid is generated by symmetric exchanges of bases. A related two-base formulation asks whether complementary basis pairs connect through such exchanges.

Why it matters and what counts as progress

Why it matters: A disconnection would obstruct exchange-based generation even though the underlying pairs have the same combined elements.

What counts as progress: A generation proof for a new class or a matroid with rigorously disconnected complementary-pair exchange classes, distinct from failure only of a stronger Gröbner-basis claim.

How the problem developed

2 Jul 2026 · Counterexample [2][1]
Matt Larson; ChatGPT 5.5 Pro found the example, and Larson proved it Gave a rank-nine binary matroid on 18 elements whose symmetric-exchange binomials do not generate its toric ideal; Larson supplies a proof without computer assistance.

AI contribution

Counterexamples to two conjectures about matroids

Matt Larson

Larson gives a rank-nine binary matroid on 18 elements whose complementary basis pairs occupy disconnected symmetric-exchange classes. It refutes Farber’s two-base conjecture and White’s toric-ideal symmetric-exchange generation conjecture.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull