Existence of 3‐chromatic Steiner quadruple systems

A Steiner quadruple system of order v (briefly SQS ( v )) is a pair ( X , $\cal B$ ), where X is a v ‐element set and $\cal B$ 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 $\cal B$ . The chromatic number of an SQS( v )( X , $\cal B$ ) is the smallest m for which there is a map $\varphi : X \rightarrow Z_m$ such that $|\varphi(B)|\geq 2$ for all $B \in \cal B$ , where $\varphi (B) =\{\varphi (x):x\in B\}$ . The system ( X , $\cal B$ ) is equitably m ‐chromatic if there is a proper coloring $\varphi$ with minimal m for which the numbers $|\varphi^{-1}(c)|, c\in Z_m$ 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