Vector-space Erdős Matching Conjecture

Let $V$ be an $n$-dimensional vector space over $\mathbb{F}_q$, and let $m_q(n,k,s)$ be the maximum size of a family $\mathcal{F}\subseteq\genfrac{[}{]}{0pt}{}{V}{k}$ containing no $s+1$ members whose sum is direct; equivalently, if $\nu_q(\mathcal{F})$ is the largest integer $s$ for which some $F_1,\ldots,F_s\in\mathcal{F}$ satisfy $\dim(F_1+\cdots+F_s)=sk$, then $\nu_q(\mathcal{F})\le s$. Vector-space Erdős Matching Conjecture. For all $n\ge (s+1)k$,

$$m_q(n,k,s)=\max\left\{\genfrac{[}{]}{0pt}{}{(s+1)k-1}{k}_q,~\genfrac{[}{]}{0pt}{}{n}{k}_q-q^{sk}\genfrac{[}{]}{0pt}{}{n-s}{k}_q\right\}.$$

The two terms are attained by the families of $k$-subspaces contained in a fixed $((s+1)k-1)$-subspace and of $k$-subspaces meeting a fixed $s$-subspace nontrivially, respectively. This is the vector-space analogue of the Erdős Matching Conjecture and connects extremal matching problems with matroid theory, coding theory, and cover-free families. The source does not state a resolution in general.