Given a family [Fscr ] of r-graphs, let ex(n, [Fscr ]) be the maximum number of edges in an n-vertex
r-graph containing no member of [Fscr ]. Let C(r)4 denote the family of r-graphs with distinct
edges A, B, C, D, such that A
∩
B = C
∩
D = Ø
and A ∪ B = C ∪ D. For s1 [les ] … [les ]
sr, let
K(r) (s1,…,sr) be the complete r-partite r-graph with parts of sizes s1,…,sr.
Füredi conjectured over 15 years ago that ex(n,C(3)4) [les ]
(n2) for sufficiently large n. We prove the weaker result
ex(n, {C(3)4, K(3)(1,2,4)}) [les ] (n2).
Generalizing a well-known conjecture for the Turán number of bipartite graphs, we
conjecture that
ex(n, K(r)(s1,…,sr)) = Θ(nr−1/s),
where s = Πr−1i=1si.
We prove this conjecture when s1 = … = sr−2 = 1 and
(i) sr−1 = 2, (ii) sr−1 = sr = 3, (iii)sr > (sr−1−1)!.
In cases (i) and (ii), we determine the asymptotic value of ex(n,K(r)(s1,…,sr)).
We also provide an explicit construction giving
ex(n,K(3)(2,2,3)) > (1/6−o(1))n8/3.
This improves upon the previous best lower bound of Ω(n29/11) obtained by probabilistic
methods. Several related open problems are also presented.