The partition characterization of relaxed SLMFs

Let $[n]=\{1,\ldots,n\}$, let $[m]=\{1,\ldots,m\}$, and write

$$\Omega=\bigcup_{j\in[n]}\omega_j\times\{j\},$$

where each $\omega_j\subseteq[m]$. For $\mathcal J\subseteq[n]$, set

$$\Omega_{\mathcal J}=\bigcup_{j\in\mathcal J}\omega_j\times\{j\}.$$

For positive integers $\nu$ and $r$, call $\Omega_{\mathcal J}$ a relaxed $(\nu,r,m)$-SLMF if

$$\sum_{j\in\mathcal J}\max\{\#(\omega_j\cap\mathcal I)-r,0\}\leq \nu(\#\mathcal I-r)$$

for every $\mathcal I\subseteq[m]$ with $\#\mathcal I\geq r+1$, with equality when $\mathcal I=[m]$. Let $G_\Omega$ be the bipartite graph associated with $\Omega$.

The partition characterization of relaxed SLMFs. Suppose $\#\Omega=r(m+n-r)$ and every vertex of $G_\Omega$ has degree at least $r+1$. Then $\Omega$ is a relaxed $(r,r,m)$-SLMF if and only if there is a partition

$$[n]=\bigcup_{\ell\in[r]}\mathcal J_\ell$$

with $\Omega_{\mathcal J_\ell}$ a relaxed $(1,r,m)$-SLMF for every $\ell\in[r]$.

The conjecture would give a purely combinatorial characterization of the relaxed SLMFs that occur in the algebraic matroid of the determinantal variety. Together with the stated sufficient and necessary conditions for a set to be a matroid base, it would complete the characterization of that algebraic matroid.