On the Number of Edges of Fan-Crossing Free Graphs

A graph drawn in the plane with $$n$$n vertices is $$k$$k-fan-crossing free for $$k \geqslant 2$$k⩾2 if there are no $$k+1$$k+1 edges $$g,e_1,\ldots , e_k$$g,e1,…,ek, such that $$e_1,e_2,\ldots ,e_k$$e1,e2,…,ek have a common endpoint and $$g$$g crosses all $$e_i$$ei. We prove a tight bound of $$4n-8$$4n-8 on the maximum number of edges of a $$2$$2-fan-crossing free graph, and a tight $$4n-9$$4n-9 bound for a straight-edge drawing. For $$k \geqslant 3$$k⩾3, we prove an upper bound of $$3(k-1)(n-2)$$3(k-1)(n-2) edges. We also discuss generalizations to monotone graph properties.