Many touchings force many crossings

Given n continuous open curves in the plane, we say that a pair is touching if they have finitely many interior points in common and at these points the first curve does not get from one side of the second curve to its other side. Otherwise, if the two curves intersect, they are said to form a crossing pair. Let t and c denote the number of touching pairs and crossing pairs, respectively. We prove that c >= 1/0(5) t(2)/n(2), provided that t >= 10n. Apart from the values of the constants, this result is best possible. (C) 2018 Elsevier Inc. All rights reserved.