The triangle chromatic index of Steiner triple systems (Q2712525)

The authors generalize the concept of chromatic index of a Steiner triple system \(S\) as follows. Let \(C\) be a given substructure, e.g. two interseting lines or two parallel lines or a triangle. An admissible colouring is a colouring of the lines such that there is no monochromatic substructure \(C\). The chromatic index \(\chi(C,S)\) is the minimum number of colours for an admissible colouring. If \(C\) consists of two intersecting lines we get the ordinary chromatic index. In the present paper \(C\) is a triangle, i.e. three lines with three distinct intersection points. \(\chi(C,v)\) denotes the minimum of all \(\chi(C,S)\) over all Steiner triple systems with \(v\) points. A cap is a subset of points with no three on a line. \(A(n)\) and \(P(n)\) denote the sizes of largest caps in the affine geometry \(\text{AG}(3, 3)\) and the projective geometry \(\text{PG}(3, 3)\), respectively, and the minimum number of caps required to partition \(\text{AG}(3, 3)\) and \(\text{PG}(3, 3)\) is denoted by \(\alpha(n)\) and \(\pi(n)\).NEWLINENEWLINENEWLINEThe authors obtain several results on these numbers as well as on the chromatic index \(\chi(C,v)\), where \(C\) is a triangle. They get exact values for the small Steiner systems with \(v= 7,9,13\), and 15, and upper bounds for larger values of \(v\), in the order of magnitude of the square root of \(v\). In concluding remarks they give their opinion that their results are unlikely to be best possible and that more work should be done.