$\{ k,n \}$-archi di un piano grafico finito, con particolare riguardo a quelli con due caratteri. Nota I

Tallini Scafati, Maria

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 40 (1966) no. 5, pp. 812-818

Résumé

The present paper, divided into Notes I and II, deals with $\{ k,n \}$-arcs in a finite graphic plane $\pi_{q}$, i.e., with sets of $k$ point of $\pi_{q}$ such that $n + 1$ distinct points of the set are never collinear, but there are $n$ distinct points of the set which are collinear. In this Note I we establish some results for arbitrary $\{ k,n \}$-arcs, we obtain a lower bound for $k$ (which cannot be improved) in the case of $\{ k,n \}$-arcs which are complete (i.e., not contained in a $\{ k+1,n \}$-arc), and we add some properties holding for arcs which only admit $m$-secant and $n$-secant lines ($m n, m \neq 0$).