Geodesic
On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009).

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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. 
@article{DMTCS_2009_special_256_a23,
     author = {Vinh, Le Anh},
     title = {On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)},
     year = {2009},
     doi = {10.46298/dmtcs.2701},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2701/}
}
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%A Vinh, Le Anh
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%J Discrete mathematics & theoretical computer science
%D 2009
%V DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)
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Vinh, Le Anh. On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009). doi : 10.46298/dmtcs.2701. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2701/

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