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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.
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Le Anh Vinh. On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.871-880. ⟨hal-01185393⟩

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