Extremal problems on set systems

• Authors: Peter Frankl;Vojtech Rödl
• Venue: Random Structures & Algorithms
• Year: 2002

For a family F(k) = {F1(k), F2(k), ..., Ft(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 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 F(k) = {F1(k), F2(k)} 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.