Massimo Giulietti - Arcs in Desarguesian nets

Journal article

A. Beato, G. Faina, M. Giulietti
Contributions Discret. Math., 2008

Cite

APA Click to copy
Beato, A., Faina, G., & Giulietti, M. (2008). Arcs in Desarguesian nets. Contributions Discret. Math.

Chicago/Turabian Click to copy
Beato, A., G. Faina, and M. Giulietti. “Arcs in Desarguesian Nets.” Contributions Discret. Math. (2008).

MLA Click to copy
Beato, A., et al. “Arcs in Desarguesian Nets.” Contributions Discret. Math., 2008.

BibTeX Click to copy

@article{a2008a,
  title = {Arcs in Desarguesian nets},
  year = {2008},
  journal = {Contributions Discret. Math.},
  author = {Beato, A. and Faina, G. and Giulietti, M.}
}

Abstract

A trivial upper bound on the size k of an arc in an r-net is $k \leq r + 1$. It has been known for about 20 years that if the r-net is Desarguesian and has odd order, then the case $k = r + 1$ cannot occur, and $k \geq r - 1$ implies that the arc is contained in a conic. In this paper, we show that actually the same must hold provided that the difference $r - k$ does not exceed $\sqrt{k/18}$. Moreover, it is proved that the same assumption ensures that the arc can be extended to an oval of the net.