Finite linear spaces in which any n-gon is Euclidean (Q1079187)

Let \({\mathcal L}\) be a linear space. A n-gon S of \({\mathcal L}\) is a set \(\{p_ i\}_{i=1,...,n}\) of points of \({\mathcal L}\), no three of which are collinear. A diagonal of S is a line \(p_ ip_ j\). A n-gon (n\(\geq 5)\) is called Euclidean if it contains at most one 4-gon having two disjoint diagonals. The authors prove that if \({\mathcal L}\) is finite and has lines of at least four points and if any 5-gon of \({\mathcal L}\) is Euclidean, then \({\mathcal L}\) is a projective space or a punctured projective space. Furthermore, if lines has at least three points and if any n-gon with \(n\geq 6\) is Euclidean, then \({\mathcal L}\) is a projective space.