In a Steiner triple system of order v, STS(v), a set of three lines intersecting pairwise in three distinct points is called a triangle. A set of lines containing no triangle is called triangle-free. The minimum number of triangle-free sets required to partition the lines of a Steiner triple system S, is called the triangle chromatic index of S. We prove that for all admissible v, there exists an STS (v) with triangle chromatic index at most 8√3v. In addition, by showing that the projective geometry PG(n,3) may be partitioned into O(6n/5) caps, we prove that the STS(v) formed the points and lines of the affine geometry AG(n,3) has triangle chromatic index at most Avs, where s=log6/(3log5)≈0.326186, and A is a constant. We also determine the values of the index for STS(v) with v≤13
