Extremal problems on set systems

For a family F(k) = {F-1((k)), F-2((k)),..., F-f((k))} 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 r(k)(n) denote the maximum cardinality of a set of integers Z subset of [n], where Z contains no arithmetic progression of length k. For any k greater than or equal to 3 we introduce families F(k) = {F-1((k)), F-2((k))} and prove that n(k-2) r(k)(n) less than or equal to ex(nk(2), F(k)) less than or equal to c(k)n(k-1) holds. We conjecture that ex (n, F(k)) = o(n(k-1)) holds. If true, this would imply a celebrated result of Szemeredi stating that r(k)(n) = o(n). By an earlier result o Ruzsa and Szemeredi, 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. (C) 2002 Wiley Periodicals, Inc.