Equitable colourings in the Witt designs

There is a vast literature already in existence on colourings in graphs and designs. We refer the interested reader to [2, 7, 8, 11, 12]. A major application of such colourings is to sampling and scheduling problems. For an excellent consideration of designs for statistical purposes, see [13]; in [8], examples of graph colouring applications in scheduling are described. Let P be a point set and B a set of subsets of P which we shall call blocks. A colouring of (P, B) is a partition of the point set such that no element of B is entirely contained in an element of the partition. A colouring is a blocking set if it is a partition into precisely two classes. A colouring is equitable if the partition classes are of at most two consecutive sizes. Equitable colourings of Steiner triple systems have been studied by Colbourn, Linek and Rosa [9] and by Haddad [10]. For the five Witt systems based on the Mathieu groups, a thorough analysis of blocking sets has been done by Berardi, Eugeni and Ferri [3, 4, 5, 6]. In this paper we study equitable colourings of these designs. Let S = S(t, k, v) be a design. If S has an equitable colouring in which the s elements of the partition have the same size a, we refer to this as an as-colouring. If s elements of the partition have size a, and t have size b, where s, t 2: 1, we refer to this as an aSbt-colouring. For convenience in what follows, an aSbt-colouring with t = 0 is an as-colouring. We prove the following in section 2: