Remarks on the Erd\H{o}s Matching Conjecture for Vector Spaces

Abstract

In 1965, Paul Erd\H{o}s asked about the largest family $Y$ of $k$-sets in $\{ 1, \ldots, n \}$ such that $Y$ does not contain $s+1$ pairwise disjoint sets. This problem is commonly known as the Erd\H{o}s Matching Conjecture. We investigate the $q$-analog of this question, that is we want to determine the size of a largest family $Y$ of $k$-spaces in $\mathbb{F}_q^n$ such that $Y$ does not contain $s+1$ pairwise disjoint $k$-spaces. Here we call two subspaces disjoint if they intersect trivially. Our main result is, slightly simplified, that if $16 s \leq \min\{ q^{\frac{n-k}{4}},$ $q^{\frac{n-2k+1}{3}} \}$, then $Y$ is either small or a union of intersecting families. Thus we show the Erd\H{os} Matching Conjecture for this range. The proof uses a method due to Metsch. We also discuss constructions. In particular, we show that for larger $s$, there are large examples which are close in size to a union of intersecting families, but structurally different. As an application, we discuss the close relationship between the Erd\H{o}s Matching Conjecture for vector spaces and Cameron-Liebler line classes (and their generalization to $k$-spaces), a popular topic in finite geometry for the last 30 years. More specifically, we propose the Erd\H{o}s Matching Conjecture (for vector spaces) as an interesting variation of the classical research on Cameron-Liebler line classes.