On proper colourings of hypergraphs using prescribed colours

Abstract

We consider a generalisation of the classic combinatorial problem of P. Erdős and A. Hajnal in the theory of hypergraphs to the case of prescribed colourings. We investigate the value m pr ( n, r ) equal to the minimum number of edges of a hypergraph in the class of n -uniform hypergraphs with prescribed chromatic number greater than r . We obtain a lower bound for this value which is better than the known results for r ≥ 3. Moreover, we give a sufficient conditions for existence of a prescribed r -colourability of an n -uniform hypergraph in terms of restrictions on the intersections of edges. As a corollary we obtain a new bound for the characteristic equal to the minimum number of edges of a hypergraph in the class of n -uniform simple hypergraphs (in which any two edges have at most one common vertex) with the prescribed chromatic number greater than r .