Many touchings force many crossings

Given n continuous open curves in the plane, we say that a pair is touching if they have only one interior point in common and at this point 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 ≥ 1/105 t2/n2, provided that t ≥ 10n Apart from the values of the constants, this result is best possible. © Springer International Publishing AG 2018