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.
On the volume set of point sets in vector spaces over finite fields
Possible cardinalities of the center of a graph
A note on a sumset in $\mathbb{Z}_{2k}$
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