An Erdős Matching Conjecture for Vector Spaces

2026-06-23

AI Summary

This work investigates the maximum size of a family of $k$-dimensional subspaces in an $n$-dimensional vector space over a finite field that contains no $s+1$ subspaces forming a direct sum, proposing and verifying an Erdős-type matching conjecture in this setting. Two natural constructions yield matching lower bounds, and the conjecture is fully resolved for $k=2$, for $n=(s+1)k$, and asymptotically when $n$ is sufficiently large. The proof combines Lovász’s matroid matching min-max theorem, high-dimensional Hoffman bounds, and vector space packing designs. Key contributions include establishing the first comprehensive conjectural framework for extremal problems of matching type in vector spaces, completely settling critical cases such as $k=2$, characterizing extremal structures for large $n$, and deriving a Hilton–Milner-type stability theorem together with a sharp connection to $t$-intersecting families, whose extremal numbers are determined up to lower-order terms.

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

Problem

• Erdős Matching Conjecture
• vector spaces
• extremal set theory
• direct sum
• subspace families

Innovation

• Erdős Matching Conjecture
• vector space
• stability theorem
• Hoffman bound
• t-cover-free families