JOURNAL OF COM!JINATORlAL THEORY (A) 22, 289-312 (1977) Partitions of Finite Relational and Set Systems JAROSLAV NESETBIL Charles University, Sokolovskd 83, 186 00 Praha 8, Czechoslovakia AND VOJTBCH MDL Czech Technical University, Husova 5, 110 00 Praha I, Czechoslovakia Communicated by P. Erdiis Received October 26, 1975 1. INTRODUCTION The classical theorem of Ramsey [12] states: Given k, I, m positive integers, there exists n with the following property: For every partition of Z-elementsubsets of a set with n elementsinto k classes, there exists a subset with m elementswith all its I-element subsets in one class. A number of structural extensions of this theorem and its analogs have appeared (see[2,4, 5, 71; [6] is a survey of this recent development). Roughly speaking, a theorem of Ramsey type guarantees for given (objects, structures,...) A, B the existence of an object C which contains so many “copies” of B that even by a partition of the set of all copies of A in C into a small number of parts, one cannot “destroy” all copies of B (see Section 2 for the exact formulation). The purpose of this paper is twofold: First, we present a method by means of which one can prove a theorem of Ramsey type for classes of set systems which are locally not too dense (i.e., do not contain complete subsystems). This (a problem of Erdas and others) was the original motivation for this research; to prove, for the set systems, a theorem analogous to the earlier results for graphs [9]. This generalization was more difficult than expected, and a new method had to be devised. The results extend all the previously known results concerning graphs (see[lo]). Second, (and a bit surprisingly), the new method is sufficiently strong to prove that one can partition not only edges, hyperedges, vertices, etc., but all suitable (called fundamental) subsystems, and in most cases to give the full characterization of this fact (seethe “prototype theorem” in [lo]). 289 Copyright 0 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.