More results on Ramsey - Turán Type problems

In her paper [9] the third author raised a general scheme of new problems. These problems can be considered as common generalizations of the problems treated in the classical results of Ramsey and Turan Since 1969 she and the first author have published a sequence of papers on the subjcc’i [5], [(;I. [4]. This work is a continuation of the above sequence. We are going to define the Ramsey-Turin function KT (,. .) below. Our main aim is to give ieasonable estimates for this function in some special cases. However before doing this we have to say a few words about notation. We hope ?hat in general these will be standard and self-explanatory, but we cln rtof stick to the special notation used in the earlier papers mentioned above. In what follows the letters k, l, m, n, Y, s, t de:rote non-negative integers. We set n={O, 1, . . . . n-l}. For;;C$ry sets A, B let [A]“=(XcA: !X]=ti}, [A]‘“= {XcA: IXjrf2}, etc. , let [A, B]“*m={XcdUB: pm+ =nAIXflBI=m}. For an arbitrary sequence &, . . ., k,-, the Ramsey function R&, , .., k,-,) equals to the minimal n such that for ail s-partitions [n+l]“= IJ EI of length icr r of n+l there are icr and Acn+l such that IAIzki and A is homogeneous for the partition in the class & i.e. [A]“cE,.