On the Number of Edges of Fan-Crossing Free Graphs.

A graph drawn in the plane with $$n$$ vertices is $$k$$ -fan-crossing free for $$k \geqslant 2$$ if there are no $$k+1$$ edges $$g,e_1,\ldots , e_k$$ , such that $$e_1,e_2,\ldots ,e_k$$ have a common endpoint and $$g$$ crosses all $$e_i$$ . We prove a tight bound of $$4n-8$$ on the maximum number of edges of a $$2$$ -fan-crossing free graph, and a tight $$4n-9$$ bound for a straight-edge drawing. For $$k \geqslant 3$$ , we prove an upper bound of $$3(k-1)(n-2)$$ edges. We also discuss generalizations to monotone graph properties. [ABSTRACT FROM AUTHOR]