标题： 向量空间上有限域大子集的距离图中的长路径 显示英文标题

标题： Long paths in the distance graph over large subsets of vector spaces over finite fields

摘要： 设$E \subset {\Bbb F}_q^d$为一个具有$q$个元素的有限域上的$d$维向量空间。 通过让顶点成为$E$的元素，并且当向量$x,y \in E$对应的两个顶点之间有一条边时，构造一个图，称为$E$的距离图，如果$||x-y||={(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=1$。 我们将证明，如果$E$的大小足够大，那么$E$的距离图包含长的不重叠路径和高程度的顶点。 显示英文摘要

摘要： Let $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements. Construct a graph, called the distance graph of $E$, by letting the vertices be the elements of $E$ and connect a pair of vertices corresponding to vectors $x,y \in E$ by an edge if $||x-y||={(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=1$. We shall prove that if the size of $E$ is sufficiently large, then the distance graph of $E$ contains long non-overlapping paths and vertices of high degree.