Colouring steiner quadruple systems

A Steiner quadruple system of order ν (briefly SQS(ν)) is a pair (X, B), where |X| = ν and B is a collection of 4-subsets of X, called blocks, such that each 3-subset of X is contained in a unique block of B. A SQS(ν) exists iff ν ≡ 2, 4 (mod 6) or ν = 0, 1 (the admissible integers). The chromatic number of (X, B) is the smallest m for which there is a map ϕ: X → Zm such that |ϕ(β)| ⩾ 2 for all β ϵ B. In this paper it is shown that for each m ⩾ 6 there exists νm such that for all admissible ν ⩾ νm there exists an m-chromatic SQS(ν). For m = 4, 5 the same statement is proved for admissible ν with the restriction that ν ≢ 2 (mod 12).