Ramsey properties of random graphs
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.
scientific article; zbMATH DE number 4168722