Rank-three case of the Kajitani–Ueno–Miyano cyclic orderability conjecture

The answer is yes. Fletcher (2026) proves the following divisible rank-three theorem.

Theorem. Let $M$ be a finite uniformly dense matroid of rank $3$ with

$$|E(M)|=3k.$$

Then $M$ admits a cyclic basis ordering.

Combined with van den Heuvel and Thomassé's coprime theorem, which applies when

$$\gcd(|E(M)|,r(M))=1,$$

this settles the full rank-three case. Indeed, if $3\nmid |E(M)|$, then

$$\gcd(|E(M)|,3)=1,$$

while the theorem above handles $3\mid |E(M)|$.

Outline of the divisible case

Write $|E|=3k$. Uniform density is then equivalent to

$$|X|\le k\,r(X)$$

for every $X\subseteq E$.

The proof separates according to whether $M$ has a nonempty proper tight set, meaning a set $X$ satisfying

$$|X|=k\,r(X).$$

Every such tight set is a flat and, in rank $3$, has rank $1$ or $2$.

• If a proper tight flat has rank $2$, its restriction is a uniformly dense rank-two matroid on $2k$ elements. A cyclic rank-two basis ordering can be constructed and interleaved with the remaining $k$ elements.

• If a proper tight flat has rank $1$, contraction by that flat produces a uniformly dense rank-two matroid on $2k$ elements. Again a rank-two cyclic ordering can be interleaved with the tight parallel class.

It remains to treat the strictly uniformly dense case, where no nonempty proper tight set exists.

For $k\ge3$, one constructs a basis $D$ meeting every rank-two flat of the largest size permitted by strict density, namely $2k-1$. Deleting this basis gives a rank-three matroid

$$M\setminus D$$

on $3(k-1)$ elements that is uniformly dense with parameter $k-1$.

Induction therefore gives a cyclic basis ordering of $M\setminus D$.

The final ingredient is a rank-three two-gap insertion theorem: if $D$ is a basis disjoint from five consecutive elements

$$p,a,b,c,q$$

for which

$$\{p,a,b\},\qquad \{a,b,c\},\qquad \{b,c,q\}$$

are bases. Then the elements of $D$ can be ordered and inserted contiguously into at least one of the two gaps $a|b$ and $b|c$, with every new length-three window forming a basis.

Since the cyclic ordering of $M\setminus D$ has $3(k-1)\ge6$ elements, this theorem applies and allows $D$ to be reinserted, completing the induction.

This insertion theorem extends McGuinness's rank-three two-gap argument for paving matroids to arbitrary rank-three matroids.

The cases $k=1$ and $k=2$ are handled separately.

This proves that every uniformly dense rank-three matroid on $3k$ elements is cyclically orderable, and hence, together with the coprime theorem of van den Heuvel and Thomassé, establishes that every finite uniformly dense rank-three matroid is cyclically orderable.

The divisible theorem has additionally been formalized in Lean 4.

Reference:
A. Fletcher, Cyclic basis orderings of uniformly dense rank-three matroids (2026).
https://doi.org/10.5281/zenodo.21813715