Existence and colouring of configurations with line size

For k = 4 it is conjectured that all configurations ( v r , b k ) exist for which the obvious necessary conditions hold. These configurations are equivalent to certain linear regular 4-uniform hypergraphs. Colouring of a configuration means to colour the points with a certain set of colours such that certain requirements for the lines are fulfilled. For k = 4 there are 5 different types of lines which will be discussed.