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geometry; polygon; intersection; combinatorial complexity
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 $kl-\left\lceil k/6\right\rceil-l$ and obtain exact bounds for $k=5$ $(f(5,l)=4l-2)$ in this case.
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