Existence of 3‐chromatic Steiner quadruple systems

A Steiner quadruple system of order v (briefly SQS (v)) is a pair (X, ), where X is a v‐element set and is a set of 4‐element subsets of X (called blocks or quadruples), such that each 3‐element subset of X is contained in a unique block of . The chromatic number of an SQS(v)(X, ) is the smallest m for which there is a map such that for all , where . The system (X, ) is equitably m‐chromatic if there is a proper coloring with minimal m for which the numbers differ from each other by at most 1. Linek and Mendelsohn showed that an equitably 3‐chromatic SQS(v) exists for v ≡ 4, 8, 10 (mod 12), v ≥ 16. In this article we show that an equitably 3‐chromatic SQS(v) exists for v ≡ 2 (mod 12) with v > 2. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 469–477, 2007