The maximum size of hypergraphs without generalized 4-cycles

Let f r ( n ) be the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain four distinct edges A, B, C, D with A ∪ B = C ∪ D and A ∩ B = C ∩ D = ∅ . This problem was stated by Erdős [P. Erdős, Problems and results in combinatorial analysis, Congr. Numer. 19 (1977) 3–12]. It can be viewed as a generalization of the Turán problem for the 4-cycle to hypergraphs. r = lim sup n → ∞ f r ( n ) / ( n r − 1 ) . Füredi [Z. Füredi, Hypergraphs in which all disjoint pairs have distinct unions, Combinatorica 4 (1984) 161–168] observed that ϕ r ⩾ 1 and conjectured that this is equality for every r ⩾ 3 . The best known upper bound ϕ r ⩽ 3 was proved by Mubayi and Verstraëte [D. Mubayi, J. Verstraëte, A hypergraph extension of the bipartite Turán problem, J. Combin. Theory Ser. A 106 (2004) 237–253]. Here we improve this bound. Namely, we show that ϕ r ⩽ min ( 7 / 4 , 1 + 2 / r ) for every r ⩾ 3 , and ϕ 3 ⩽ 13 / 9 . In particular, it follows that ϕ r → 1 as r → ∞ .