For a family \documentclass{article}\pagestyle{empty}\begin{document}$\boldmath{F}(k) = \{{\mathcal F}_1^{(k)}, {\mathcal F}_2^{(k)},\ldots,{\mathcal F}_t^{(k)}\}$\end{document} of k‐uniform hypergraphs let ex (n,F(k)) denote the maximum number of k‐tuples which a k‐uniform hypergraph on n vertices may have, while not containing any member of F(k). Let rk(n) denote the maximum cardinality of a set of integers Z⊂[n], where Z contains no arithmetic progression of length k. For any k≥3 we introduce families \documentclass{article}\pagestyle{empty}\begin{document}$\boldmath{F}(k) = \{{\mathcal F}_1^{(k)},{\mathcal F}_2^{(k)}\}$\end{document} and prove that nk−2rk(n)≤ex (nk2,F(k))≤cknk−1 holds. We conjecture that ex(n,F(k))=o(nk−1) holds. If true, this would imply a celebrated result of Szemerédi stating that rk(n)=o(n). By an earlier result o Ruzsa and Szemerédi, our conjecture is known to be true for k=3. The main objective of this article is to verify the conjecture for k=4. We also consider some related problems. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 131–164, 2002.
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