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On the number of intersections of two polygons. (English) Zbl 1099.52004
The maximum possible number $$f(k,l)$$ of intersections of a simple $$k$$-gon and $$l$$-gon lying in the same plane is known for $$k\geq 3$$, $$l\geq 3$$ if at least one of the numbers $$k,l$$ is even. Namely, $$f(k,l)=kl$$ if the both numbers are even and $$f(k,l)=k(l-1)$$ if $$k$$ is even and $$l$$ odd. However, if both $$k$$ and $$l$$ are odd then only the bounds $$l(k-1)+(3-k)\leq f(k,l)\leq l(k-1)$$ are known (for $$k\leq l$$).
The following results are proved in the paper for both $$k\leq l$$ being odd:
i) an improved upper bound of $$f(k,l)$$ is $$l(k-1)-\lceil \frac {k}{6}\rceil$$ for $$k\geq 7$$,
ii) if $$l\geq 5$$ is odd then $$f(5,l)=4l-2$$.
Reviewer: Ivan Saxl (Praha)
##### MSC:
 52C10 Erdős problems and related topics of discrete geometry 52C45 Combinatorial complexity of geometric structures
##### Keywords:
combinatorial complexity
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