The Esther-Klein-Problem in the Projective Plane

Let \(p(k)\) (\(q(k)\)) be the smallest number such that in the projective plane, every simple arrangement of \(n \geq p(k)\) (\( \geq q(k)\)) 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 \(q(k) (\geq p(k))\) is proved.