Łuczak, Tomasz; Ruciński, Andrzej; Voigt, Bernd Ramsey properties of random graphs. (English) Zbl 0711.05041 J. Comb. Theory, Ser. B 56, No. 1, 55-68 (1992). Let \(F\to (G)^ v_ r[F\to (G)^ e_ r]\) mean that for every r- coloring of the vertices [edges] of graph F there is a monochromatic copy of G in F. A rational d is said to be crucial for property \({\mathcal A}\) if for some constants c and C the probability that the binomial random graph K(n,p) has \({\mathcal A}\) tends to 0 when \(np^ d<c\) and tends to 1 while \(np^ d>C\), \(p=p(n)\), \(n\to \infty\). Let \(| G|\) and e(G) stand for the number of the vertices and edges of a graph G, respectively. We prove that \(\max_{H\subseteq G}e(H)/| H| -1\) is crucial for \(K(n,p)\to (G)^ v_ r\), whereas 2 is crucial for \(K(n,p)\to (K_ 3)^ e_ 2\). The existence of sparse Ramsey graphs is also deduced. Reviewer: A.Ruciński Cited in 3 ReviewsCited in 28 Documents MSC: 05C80 Random graphs (graph-theoretic aspects) 05C55 Generalized Ramsey theory Keywords:binomial random graph; sparse Ramsey graphs PDFBibTeX XMLCite \textit{T. Łuczak} et al., J. Comb. Theory, Ser. B 56, No. 1, 55--68 (1991; Zbl 0711.05041) Full Text: DOI