Arcs in Laguerre planes (Q2717059)

A Laguerre plane \(\mathcal L\) is a known incidence structure \((\mathcal P,\mathcal B,\|, \mathcal I)\) consisting of a point-set \(\mathcal P\), a set \(\mathcal B\) of at least two circles, an incidence relation \(\mathcal I\subset\mathcal P\times\mathcal B\) and an equivalence relation \(\|\) on \(\mathcal P\) with definite properties. \(K\subset \mathcal P\) is called a \(k\)-arc of \(\mathcal L\) if \(K\) is a set of \(k\) \((k\geq 3)\) independent points and every circle contains at most three points of it. \(K\) is said to be complete if not properly contained in a \((k+1)\)-arc.NEWLINENEWLINEThe authors investigate in Laguerre planes of order \(q\) the \(k\)-arcs with particular attention to problems of existence and completeness and obtain different interesting results.