In 1965, Paul Erdős asked about the largest family $Y$ of $k$-sets in $\{1,\dots ,n\}$ such that $Y$ does not contain $s+1$ pairwise disjoint sets. This problem is commonly known as the Erdős 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 $\le min\{q^{\frac{n-k}{4}},q^{\frac{n-2k+1}{3}}\}$, then $Y$ is either small or a union of intersecting families. Thus we show the Erdős 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 Erdős 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 Erdős Matching Conjecture (for vector spaces) as an interesting variation of the classical research on Cameron–Liebler line classes.

1965年，PaulErdős询问了最大的家庭 $ÿ$ 的 $ķ$设置在 $\{\mathrm{1个}，\dots ，ñ\}$ 这样 $ÿ$ 不含 $s+\mathrm{1个}$成对不相交集。这个问题通常被称为Erdős匹配猜想。我们调查$q$这个问题的模拟，即我们要确定最大家庭的规模 $ÿ$ 的 $ķ$中的空格 ${\mathbb{F}}_q^{ñ}$ 这样 $ÿ$ 不含 $s+\mathrm{1个}$ 成对不相交 $ķ$-空间。在这里，我们称两个子空间不相交，如果它们相交。

我们的主要结果是，稍微简化一下，即16秒 $\le 分\{q^{\frac{ñ-ķ}{4}}，q^{\frac{ñ-2ķ+\mathrm{1个}}{3}}\}$， 然后 $ÿ$是小的家庭还是相交家庭的联合。因此，我们显示了该范围的Erdős匹配猜想。证明使用了Metsch的方法。我们还将讨论构造。特别是，我们表明$s$，有很多例子，其大小与相交的家庭相近，但结构上却有所不同。

作为应用，我们讨论了向量空间的Erdős匹配猜想与Cameron-Liebler线类之间的紧密关系（以及它们对 $ķ$-spaces），这是过去30年来流行的有限几何主题。更具体地说，我们提出了对向量空间的Erdős匹配猜想，作为对Cameron-Liebler线类的经典研究的有趣变体。