Bounds for arcs of arbitrary degree in finite Desarguesian planes (Q2804931)

A \((k,n)\)-arc in the projective plane \(\mathrm{PG}(2,q)\) is a set of \(k\) points such that no \(n+1\) of the points are collinear but some \(n\) of them are collinear. The authors show that a complete \((k,n)\) arc in \(\mathrm{PG}(2,q)\) with \(n \geq 2\) and \(q \geq n\) must have \(k \geq \sqrt{n(n-1)(q+1)}\). Additionally, they show that a complete \((k,3)\)-arc in \(\mathrm{PG}(2,q)\) with \(k > (q+2)\) has that every point of the arc is incident with at least one trisecant and if the arc is complete then its trisecants induce a blocking set in the dual plane. Finally, it is shown that for \(q \geq 17\), plane \((k,3)\)-arcs with a certain incidence condition do not attain the best known upper bound.