On balanced incomplete block designs with specified weak chromatic number

A weak c-colouring of a balanced incomplete block design (BIBD) is a colouring of the points of the design with c colours in such a way that no block of the design has all of its vertices receive the same colour. A BIBD is said to be weakly c-chromatic if c   is the smallest number of colours with which the design can be weakly coloured. In this paper we show that for all c⩾2c⩾2 and k⩾3k⩾3 with (c,k)≠(2,3)(c,k)≠(2,3), the obvious necessary conditions for the existence of a (v,k,λ)(v,k,λ)-BIBD are asymptotically sufficient for the existence of a weakly c  -chromatic (v,k,λ)(v,k,λ)-BIBD.