Arcs in Desarguesian nets

Annalisa Beato, Giorgio Faina, Massimo Giulietti

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.

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PID: http://hdl.handle.net/10515/sy5w08wx1


Contributions to Discrete Mathematics. ISSN: 1715-0868