List colouring of two matroids

Definitions and mathematical statement

Let $M_1=(S,\mathcal I_1)$ and $M_2=(S,\mathcal I_2)$ be two finite matroids on the same ground set. Each independent-set family is hereditary and obeys exchange: if independent sets $I,J$ satisfy $|I|<|J|$, some element of $J\setminus I$ can be added to $I$ while preserving independence. A subset is common independent if it belongs to both families.

Suppose $S$ has a partition into $k$ common independent sets, where $k\ge1$ and empty parts are permitted. Give each $s\in S$ an arbitrary list $L(s)$ of $k$ distinct colours. Must there be a map $f$ with \[ f(s)\in L(s)\quad(s\in S),\qquad f^{-1}(a)\in\mathcal I_1\cap\mathcal I_2 \quad\text{for every colour }a? \] The available colours need not come from a common palette of size $k$. Only the size of each individual list is prescribed.

The same colour classes must be independent in both matroids. Solving two list-colouring problems separately would not ensure this compatibility. No field representation or special matroid class is assumed. A list-colouring theorem for a single matroid therefore does not directly give the two-matroid assertion. The source's ordinary colourability hypothesis is exactly the stated partition into common independent sets, rather than separate partitions for $M_1$ and $M_2$.

Short English statement

Whenever two matroids admit a common $k$-colouring, do they admit a common colouring from every assignment of lists of size $k$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3000048 (AMR-029-0048), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Egres Open - List of open problems, Open-problem page: List colouring of two matroids. Original source indexed by AMR-029-0048. https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems.
• S3 EGRES Open, List colouring of two matroids. Individual source page and explanatory remarks. https://lemon.cs.elte.hu/egres/open/List_colouring_of_two_matroids.