On ramsey families of sets

The main result of this paper is a lemma which can be used to prove the existence of highchromatic subhypergraphs of large girth in various hypergraphs. In the last part of the paper we use amalgamation techniques to prove the existence for everyl, k 3 of a setA of integers such that the hypergraph having as edges all the arithmetic progressions of lengthk inA has both chromatic number and girth greater thanl.