Abstract
LetH be a fixed graph of chromatic numberr. It is shown that the number of graphs onn vertices and not containingH as a subgraph is\(2^{(\begin{array}{*{20}c} n \\ 2 \\ \end{array} )(1 - \frac{1}{{r - 1}} + o(1))} \). Leth r (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 thath r (n) =o(n 2) although for every fixedc < 2 one has lim n→∞ h r (n)/n c = ∞.
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Erdös, P., Frankl, P. & Rödl, V. The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs and Combinatorics 2, 113–121 (1986). https://doi.org/10.1007/BF01788085
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DOI: https://doi.org/10.1007/BF01788085