... any drawing of Km,n The cyclic ordering of the edges around each bj∈ B can be considered as a cyclic ordering of the elements of A, and therefore as an element of S∗m.Suppose we have a K2,mwhich ... a 1 ,a 2 ,a 3, ,a m,and the elements of B are b1,b2, ,bn.InadrawingD,wedenotebycrD (a i ,a k )the number of crossings of arcs, one terminating at a i, the other at a kand by crD (a i )the number ... A, andletσ have the same definition as it had in the proof of (3).Then σ ≥m5Z(5,n)+2θ.In the following lemma, A is defined in the same way as in the proof of Lemma 1.Lemma 3 Let λ be the number...