Systems of curves on surfaces
It is proved that for each compact (bordered) surface \(\Sigma\) and a positive integer \(k\) there is a constant \(N\) with the following property: If \(\Gamma\) is a family of pairwise nonhomotopic closed curves on \(\Sigma\) such that any two curves from \(\Gamma\) intersect in at most \(k\) points and each curve has at most \(k\) self-intersections, then \(\Gamma\) contains at most \(N\) curves. This result does not hold for families of curves with bounded algebraic intersection. Several examples are provided to show that the smallest possible \(N\) is rather large. The main result is used to show that for each compact surface \(\Sigma\) and each positive integer \(k\), there is only a finite number of distinct \(k\)-minimal triangulations of \(\Sigma\) and only a finite number of distinct embeddings of graphs in \(\Sigma\) that are minimal with respect to their face-width being equal to \(k\).
scientific article; zbMATH DE number 934821