The drawing Ramsey number Dr(Kn)

Bounds are determined for the smallest m = Dr(Kn) such that every drawing of Km in the plane (two edges have at most one point in common) contains at least one drawing of Kn with the maximum number (:) of crossings. For n = 5 these bounds are improved to 11 :::; Dr(K5) 113. A drawing D( G) of a graph G is a special realization of G in the plane. The vertices are mapped into different points of the plane (also called vertices of D( G)), the edges are mapped into lines (also called edges of D( G)) connecting the corresponding vertices such that two edges have at most one point in common, which is either a common vertex or a crossing. Two drawings are said to be isomorphic if there exists an incidence-preserving one-to-one correspondence between vertices, crossings, edges, parts of edges and regions. It is well known that every drawing of the complete graph K4 has at most one crossing. Thus, the maximum number of crossings in a drawing D(Kn) is at most (:). Different nonisomorphic drawings D(Kn) with (:) crossings are discussed in [4]. In this note, we will show that for m sufficiently large every drawing of D( Km) must contain at least one drawing D(Kn) with (:) crossings. Moreover, bounds for the smallest such m, denoted by Dr(Kn), will be deduced. It can be observed that the question for a sub drawing D(Kn) with maximum number of crossings is similar to the Esther Klein problem if lines are used instead of straight line segments and if convexity of n points is replaced by drawings D(Kn) with (:) crossings. Theorem 1. For every positive integer n there exists a least integer Dr(Kn) such that every drawing D(Km) with m 2 Dr(Kn) contains a subdrawing D(Kn) with (:) crossings. Australasian Journal of Combinatorics ll( 1995) I pp.151-156 Proof. The existence of Dr(Kn) will be deduced from Ramsey's theorem. Consider a drawing D(Km) with m 2: r4(5, n), where the Ramsey number r4(5, n) denotes the smallest I such that in every 2-coloring of the four-element subsets of an I-element set V, using colors green and red, there is a 5-element subset of V with a1l4-element subsets green or an n-element subset of V with all 4-element subsets red. Color a 4-element subset of the vertex set V of D( Km) red if the four vertices determine a crossing and green otherwise. Among any five vertices there are four determining a crossing, since Ks is nonplanar. Thus, there exists no 5-element subset of V with all 4-element subsets colored green, and there must be an n-element subset of V with all 4-element subsets red. These n vertices determine (~) crossings and Theorem 1 is proved. I11III The proof of Theorem 1 yields Dr(Kn) r4(5, n). This bound might be very far from the truth, since none of the topological aspects of the problem besides the non-planarity of Ks is taken into account. Moreover, in case n 2: 5 only rough upper bounds are available for r4(5, n) (see for example [3)). A lower bound for Dr(Kn) can be deduced from the Esther Klein problem. In [5,6] it was shown that for n 2: 2 there are 2n.-2 points in the plane no three of them collinear and no n of them determining a convex n-gon. Take 2n.-2 such points as vertices of a drawing of a complete gra~h and draw all edges as straight line segments. Then no subdrawing D(Kn) with \~) crossings can occur, since among any n vertices there are four forming a non-convex 4-gon and hence having no crossing. Thus we obtain Theorem 2. 2n2 + 1 :::; Dr(Kn) :::; for n 2: 2. Figure 1. A D(f{lO) containing no sub drawing D(Ks) with five crossings Trivially, Dr(Kn} = n for n :::; 3, and Theorem 2 implies Dr(K4) = 5. For n 2:: 5, no exact values of Dr(Kn) are known so far. The next theorem will improve the