Arcs in Desarguesian nets

Journal article

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

Semantic Scholar DBLP DOI


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

Beato, A., G. Faina, and M. Giulietti. “Arcs in Desarguesian Nets.” Contributions Discret. Math. (2008).

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


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.


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