## Jakub \v Cern\'y, Jan K\'ara, Daniel Kr\'al', Pavel Podbrdsk\'y, Miroslava Sot\'akov\'a, Robert \v S\'amal On the number of intersections of two polygons

Comment.Math.Univ.Carolinae 44,2 (2003) 217-228.

Abstract:We study the maximum possible number \$f(k,l)\$ of intersections of the boundaries of a simple \$k\$-gon with a simple \$l\$-gon in the plane for \$k,l\ge 3\$. To determine the number \$f(k,l)\$ is quite easy and known when \$k\$ or \$l\$ is even but still remains open for \$k\$ and \$l\$ both odd. We improve (for \$k\le l\$) the easy upper bound \$kl-l\$ to \$kl-\ceil {k/6}-l\$ and obtain exact bounds for \$k=5\$ \$(f(5,l)=4l-2)\$ in this case.

Keywords: geometry, polygon, intersection, combinatorial complexity
AMS Subject Classification: 52C45, 52C10