On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields

Abstract

We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence.

Keywords

• finite non-euclidean graphs
• spectral graphs
• distance problem
• finite euclidean graphs
• [math.math-co] mathematics [math]/combinatorics [math.co]
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]