The Esther-Klein-Problem in the Projective Plane

Abstract

Let $p(k)$ ($q(k)$) be the smallest number such that in the projective plane, every simple arrangement of $n\ge p(k)$ ($\ge q(k)$) straight lines (pseudolines) contains $k$ lines which determine a $k$-gonal region. The values $p(6)=q(6)=9$ are determined and the existence of $q(k)(\ge p(k))$ is proved.