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For $d\ge 2$ and an odd prime power $q$, consider the vector space$\mathbb{F}_q^d$ over the finite field $\mathbb{F}_q$, where the distancebetween two points $(x_1,\ldots,x_d)$ and $(y_1,\ldots,y_d)$ is defined as$\sum_{i=1}^d (x_i-y_i)^2$. A distance graph is a graph associated with anon-zero distance to each of its edges. We show that large subsets of vectorspaces over finite fields contain every nearly spanning distance tree withbounded degree in each distance. This quantitatively improves results byBennett, Chapman, Covert, Hart, Iosevich, and Pakianathan on finding distancepaths, and results by Pham, Senger, Tait, and Thu on finding distance trees. Akey ingredient in proving our main result is to obtain a colorfulgeneralization of a classical result of Haxell about finding nearly spanningbounded-degree trees in an expander.

Almost spanning distance trees in subsets of finite vector spaces