Uniquely 3-colourable Steiner triple systems

A Steiner triple system (STS(v)) is said to be 3-balanced if every 3-colouring of it is equitable; that is, if the cardinalities of the colour classes differ by at most one. A 3-colouring, φ, of an STS(v) is unique if there is no other way of 3-colouring the STS(v) except possibly by permuting the colours of φ. We show that for every admissible v⩾25, there exists a 3-balanced STS(v) with a unique 3-colouring and an STS(v) which has a unique, non-equitable 3-colouring.