Mason’s conjecture on freest matroid tensor products
For every pair of uniform matroids and , there exists a freest matroid product of and ; equivalently, the relevant class of products has a maximal element in the freeness order.
Equivalent formulations 1Other wordings
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
Generic birigidity formulation
For all admissible parameters , the generic birigidity matroid is the freest abstract -birigidity matroid.
References
Primary source
Additional references
- Linear matroid products: a synthetic approach — arXiv — Mykhaylo Tyomkyn
Progress summary
A September 2026 preprint proves the conjecture for an important representable class, but the full conjecture for arbitrary matroids remains open.
Mason’s 1981 question concerns the existence and maximality of freest tensor products of matroids. The new result handles linearly representable matroids, rather than the full class of arbitrary matroids.
September 2026 representable-case result
On September 15, 2026, Mykhaylo Tyomkyn’s preprint Linear matroid products: a synthetic approach claimed that star-basis normal forms establish the desired maximality for linearly representable abstract birigidity matroids. It also reports representation results for -rigidity and hyperconnectivity matroids, but does not claim the general conjecture.
Current status (as of September 2026): The conjecture is claimed for linearly representable matroids, while its validity for arbitrary matroids remains open.
Solutions 0
No solutions have been posted yet.