Colouring problems for symmetric configurations with block size 3

The study of symmetric configurations v3 with block size 3 has a long and rich history. In this paper we consider two colouring problems which arise naturally in the study of these structures. The first of these is weak colouring, in which no block is monochromatic; the second is strong colouring, in which every block is multichromatic. The former has been studied before in relation to blocking sets. Results are proved on the possible sizes of blocking sets and we begin the investigation of strong colourings. We also show that the known 213 and 223 configurations without a blocking set are unique and make a complete enumeration of all nonisomorphic 203 configurations. We discuss the concept of connectivity in relation to symmetric configurations and complete the determination of the spectrum of 2‐connected symmetric configurations without a blocking set. A number of open problems are presented

Blocking set free configurations and their relations to digraphs and hypergraphs

AbstractThe current state of knowledge concerning the existence of blocking set free configurations is given together with a short history of this problem which has also been dealt with in terms of digraphs without even dicycles or 3-chromatic hypergraphs. The question is extended to the case of nonsymmetric configurations (vr, b3). It is proved that for each value of r ≥ 3 there are only finitely many values of v for which the existence of a blocking set free configuration is still unknown