Log-concavity of matroid flat counts
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Must the counts W_i of rank-i flats of a matroid be log-concave, so W_i²≥W_(i−1)W_(i+1)? The focused graphic counterexample uses four paths between two vertices, three of length 26 and one of length one.
Why it matters and what counts as progress
Why it matters: It tests a proposed coefficient-shape constraint for matroid flats. Log-concavity is stronger than the separate unimodality question.
What counts as progress: An exact flat-count violation of log-concavity, or a corrected inequality for a restricted matroid class. Unimodality remains a separate target.
How the problem developed
- 1972 · Conjecture [1]
J. H. Mason Proposed log-concavity conjectures for matroid sequences, including flat counts.
- 2 Jul 2026 · Counterexample [1]
Matt Larson Gave counterexamples to Mason’s log-concavity conjecture for flat counts, including a graphic matroid with a failure near its top rank.
- 24 Jul 2026 · Follow up [2]
Alexander Divoux, Chayim Lowen and Shouda Wang Showed that suitable failures of flat-count log-concavity yield failures of unimodality, producing separate counterexamples to Rota’s unimodality conjecture.
AI contribution
Counterexamples to two conjectures about matroids
A graphic matroid formed from four paths between two vertices, three of length 26 and one of length 1, has flat counts W₇₄²=967,813,250,625 < W₇₃W₇₅=982,359,393,275. This refutes Mason’s plain log-concavity conjecture for matroid flats.
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