Thrackles: An Improved Upper Bound

A {\em thrackle} is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of $n$ vertices has at most $1.3984n$ edges. {\em Quasi-thrackles} are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an {\em odd} number of times. It is also shown that the maximum number of edges of a quasi-thrackle on $n$ vertices is ${3\over 2}(n-1)$, and that this bound is best possible for infinitely many values of $n$.