论文标题
关于矢量空间的Erds匹配猜想的评论
Remarks on the Erdős Matching Conjecture for Vector Spaces
论文作者
论文摘要
1965年，PaulErdős询问了$ \ {1，\ ldots，n \} $最大的$ y $ y $ $ y $ $ k $ set，以便$ y $不包含$ s+1 $ $ $ $ $ $。这个问题通常称为ERD匹配猜想。我们调查了这个问题的$ q $ - analog，也就是说，我们要确定$ \ mathbb {f} _q^n $中最大的$ y $ y $ y $ y $ y $ y $ y $ y $ y $不包含$ s+s+s+1 $成对的不相关$ k $ spaces。在这里，我们将两个子空间不相交，如果它们微不足道。 我们的主要结果是稍微简化的，如果$ 16 s \ leq \ min \ {q^{\ frac {n-k} {4}}}，$ $ q^{\ frac {\ frac {n-2k+1} {3}} {3}}} \} \ y $是小或$ y $的$ y $是一家人的结合。因此，我们显示了此范围的ERD \ h {OS}匹配的猜想。该证明使用METSCH引起的方法。我们还讨论结构。特别是，我们表明，对于较大的$ s $，有很多大小与相交家庭的结合，但结构上有所不同。 作为一个应用程序，我们讨论了矢量空间的ERDS匹配猜想与Cameron-Liebler Line类（及其对$ K $ spaces的概括）之间的密切关系，这是过去30年来有限几何学的流行话题。更具体地说，我们提出了ERD的匹配猜想（对于向量空间），这是对Cameron-Liebler Line类的经典研究的有趣变化。
In 1965, Paul Erdős 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 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 $16 s \leq \min\{ q^{\frac{n-k}{4}},$ $q^{\frac{n-2k+1}{3}} \}$, 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 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.
