For and an odd prime power , consider the vector space over the finite field , where the distance between two points and is defined as . A distance graph is a graph associated with a nonzero distance to each of its edges. We show that large subsets of vector spaces over finite fields contain every nearly spanning distance tree with bounded degree in each distance. This quantitatively improves results by Bennett, Chapman, Covert, Hart, Iosevich, and Pakianathan on finding distance paths, and results by Pham, Senger, Tait, and Thu on finding distance trees. A key ingredient in proving our main result is to obtain a colorful generalization of a classical result of Haxell about finding nearly spanning bounded-degree trees in an expander.

中文翻译：

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有限向量空间子集中的几乎跨越距离树

为了和一个奇素数幂，考虑向量空间在有限域上，其中两点之间的距离和定义为。距离图是与其每条边的非零距离相关联的图。我们证明了有限域上向量空间的大子集包含每个距离上具有有界度的几乎跨越距离树。这定量地改进了 Bennett、Chapman、Covert、Hart、Iosevich 和 Pakianathan 在查找距离路径方面的结果，以及 Pham、Senger、Tait 和 Thu 在查找距离树方面的结果。证明我们主要结果的一个关键因素是获得 Haxell 经典结果的丰富多彩的概括，即在扩展器中找到几乎跨越有界度树。