Let \cH be a family of k-graphs (k-uniform hypergraphs). Let ex(n,\cH) denote their Turan number, i.e., the maximum number of k-sets avoiding all members of \cH.
Very recently Bukh, Conlon and Fox showed that given any rational number r, 1< r< 2, there is a finite set of graphs \cH with \ex(n,\cH)= \Theta (nr). Similar statement was proved for k-uniform hypergraphs by Frankl 30 years ago.
Here we give a short, concise proof for the following result. There exists a k-uniform hypergaph H (for each k\geq 5) without exponent, i.e., when the Turan function is not polynomial in n. More precisely, we have \ex(n,H)=o(nk-1) but it exceeds nk-1-c for any positive c for n> n0(k,c).
This is an extension (and simplification) of a result of Frankl and the speaker from 1987 (where the case k=5 was proven). We conjecture that it is true for all k\geq 4 (and probably for k=3 as well).
Minden érdeklődőt szeretettel várunk,
Péter