J. Korean Math. Soc. 2019; 56(6): 1515-1528
Online first article July 17, 2019 Printed November 1, 2019
https://doi.org/10.4134/JKMS.j180776
Copyright © The Korean Mathematical Society.
Alex Iosevich, Hans Parshall
University of Rochester; The Ohio State University
We show that large subsets of vector spaces over finite fields determine certain point configurations with prescribed distance structure. More specifically, we consider the complete graph with vertices as the points of $A \subseteq \mathbf{F}_q^d$ and edges assigned the algebraic distance between pairs of vertices. We prove nontrivial results on locating specified subgraphs of maximum vertex degree at most $t$ in dimensions $d \geq 2t$.
Keywords: Erd\H{o}s distance problem, finite fields, graph theory
MSC numbers: 52C10, 11T23
2016; 53(1): 115-126
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