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#This author is partially supported by NSF grant DMS-0457512. 1 Also, we observe that the demonstrated non-principality phenomenon holds also with respect to the Ramsey-Tur'an density as well. 1 Introduction In this paper, we prove the non-principality phenomenon for the classical extremal problems for k-uniform hypergraphs. The main motivation is to study the qualitative difference between the cases k = 2, and k> = 3, and our results for the Tur'an problem exhibit this difference. We also study this question in the context of Ramsey-Tur'an theory, introduced by Erd&quot;&quot;os and S'os. Although we prove the non-principality phenomenon for Ramsey-Tur'an problems when k> = 3, the behavior for k = 2 remains open. This is one of the few cases where an extremal problem for hypergraphs can be solved but not for graphs. Given a family F of k-uniform hypergraphs (k-graphs for short), le

Year: 2008

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