An Erdős Matching Conjecture for Vector Spaces

Abstract

We study a vector-space analogue of the Erdős Matching Conjecture. Let $m_q(n,k,s)$ denote the maximum cardinality of a family of $k$-dimensional subspaces of an $n$-dimensional vector space over ${\mathbb{F}}_q$ with no $s+1$ members whose sum is direct. Two natural constructions provide lower bounds. The first consists of all $k$-subspaces contained in a fixed $((s+1)k-1)$-dimensional subspace; the second consists of all $k$-subspaces that intersect a fixed $s$-dimensional subspace nontrivially. These constructions motivate the following vector-space analogue of the Erdős Matching Conjecture: for all $n\ge (s+1)k$, $$m_q(n,k,s)=\max \left\{{\left[\genfrac{}{}{0ex}{}{(s+1)k-1}{k}\right]}_q,{\left[\genfrac{}{}{0ex}{}{n}{k}\right]}_q-q^{ks}{\left[\genfrac{}{}{0ex}{}{n-s}{k}\right]}_q\right\}.$$ We prove this conjecture when $k=2$, when $n=(s+1)k$, and when $n$ is sufficiently large. In particular, the case $k=2$ may be viewed as a vector-space analogue of the Erdős--Gallai theorem. In the large-$n$ range, we also prove a Hilton--Milner-type stability theorem, determining the largest nontrivial families with this property. Finally, we connect this problem with $t$-cover-free families in vector spaces and determine their extremal number up to a lower-order term, extending a recent result of Shan and Zhou for the special case $t=2$. The proofs combine Lovász's minimax theorem for matroid matchings, a high-dimensional Hoffman bound for uniform hypergraphs, and packing-design arguments in vector spaces.