Upper bound on common independent set cover

Definitions and mathematical statement

For a finite loopless matroid $M=(S,r)$ on a nonempty ground set, $r(X)$ is the maximum size of an independent subset of $X$. Independence is hereditary and satisfies the exchange axiom. Looplessness ensures $r(X)>0$ for every nonempty $X$. Define the density parameter \[ \Delta(M)= \max_{\varnothing\ne X\subseteq S}\frac{|X|}{r(X)}. \] The empty set is omitted to avoid the meaningless quotient $0/0$.

Let $M_1=(S,r_1)$ and $M_2=(S,r_2)$ be two loopless matroids on the same ground set, and put $q=\lceil\max\{\Delta(M_1),\Delta(M_2)\}\rceil+1$. Must there be a partition $S=I_1\sqcup\cdots\sqcup I_q$ with \[ r_1(I_j)=r_2(I_j)=|I_j| \quad(1\le j\le q)? \] Empty classes are permitted, so this means a cover or partition using at most $q$ common independent sets.

Each density supplies a necessary lower bound for the number of independent classes for its own matroid. The conjecture asks whether compatibility of the two requirements costs at most one additional class above the larger rounded density. Separate partitions for the two matroids do not provide the requested common partition. The assertion places no representability or special structural restriction on either matroid.

Short English statement

Can two loopless matroids always be simultaneously coloured using at most one more colour than their larger rounded density requires?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3000086 (AMR-029-0086), 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: Upper bound on common independent set cover. Original source indexed by AMR-029-0086. https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems.
• S3 EGRES Open, Upper bound on common independent set cover. Individual source page and explanatory remarks. https://lemon.cs.elte.hu/egres/open/Upper_bound_on_common_independent_set_cover.