Different levels of randomness in Random Ramsey theorems

Extremal graph theory, Ramsey theory and the theory of Random graphs are strongly connected to each other. ng from these fields, we formulate some problems and results which are related to different levels of randomness. The first one is completely non-random, being the ordinary Ramsey-Turán problem and in the subsequent three problems we formulate some randomized variations of the problem. ng of graph properties, we shall consider them as sets of graphs and occasionally write G ∈ P instead of writing that G has property P. A graph property P is called monotone if adding an edge to a Hn ∈ P, we get an H∗n ∈ P. ll use three models of random graphs: the binomial, the hypergeometric and the stopping-rule model. bstract contains the most important definitions.