Upper chromatic number of Steiner triple and quadruple systems

Abstract

The upper chromatic number χ( H) of a set system H is the maximum number of colours that can be assigned to the elements of the underlying set of H in such a way that each H ϵ H contains a monochromatic pair of elements. We prove that a Steiner triple system of order v ⩽ 2 k − 1 has an upper chromatic number which is at most k. This bound is the best possible, and the extremal configurations attaining equality can be characterized. Some consequences for Steiner quadruple systems are also obtained.