What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even: If both m and n are even, then every pair of sides may cross and so the answer is mn. If exactly one polygon, say the n-gon, has an odd number of sides, it can intersect each side of the m-gon polygon at most n−1 times; hence there are at most mn−m intersections. It is not hard to construct examples that meet these bounds. If both m and n are odd, the best known construction has mn−(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn−(m+⌈n/6⌉), for m≥n. We prove a new upper bound of mn−(m+n)+C for some constant C, which is optimal apart from the value of C.
