Circle spaces in a linear space (Q5954174)

A circle space is a linear space \({\mathcal S}\) together with certain subsets of the point set of \({\mathcal S}\) called circles such that through three noncollinear points of \({\mathcal S}\) there goes a unique circle. In this article the authors investigate circle spaces for which the circles are either all \(m\)-arcs, \(m\geq 3\), or all Fano planes. A \({\mathcal C}_m\) space is a circle space in which every circle is an \(m\)-arc, \(m\geq 3\). The authors prove that in every finite affine plane there is a \({\mathcal C}_4\) space. They also discuss the existence of \({\mathcal C}_m\) spaces in finite projective planes, and in particular, they prove the existence of a \({\mathcal C}_m\) space with \(m>3\) in a projective plane of prime order \(p\) implies \(p=2\) and \(m=4\). Also, if a \({\mathcal C}_m\) space with \(m\) even exists in a finite projective plane of order \(n\), then \(n\) must be even. This last result is used to prove that a circle space with circles Fano planes can not exist in a finite projective plane of even order.