The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent

AbstractLetH be a fixed graph of chromatic numberr. It is shown that the number of graphs onn vertices and not containingH as a subgraph is $$2^{(\begin{array}{*{20}c} n \\ 2 \\ \end{array} )(1 - \frac{1}{{r - 1}} + o(1))} $$ . Lethr(n) denote the maximum number of edges in anr-uniform hypergraph onn vertices and in which the union of any three edges has size greater than 3r − 3. It is shown thathr(n) =o(n2) although for every fixedc < 2 one has limn→∞hr(n)/nc = ∞.