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 ⩾ 2 and k ⩾ 3 with ( c , k ) ≠ ( 2 , 3 ) , the obvious necessary conditions for the existence of a ( v , k , λ ) -BIBD are asymptotically sufficient for the existence of a weakly c-chromatic ( v , k , λ ) -BIBD.