Graph Theory: Ramsey Numbers, Markov's Inequality, Expected Cycles, Chromatic Number, Slides of Design Patterns

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For every integer k ≥ 3 , the Ramsey number of k satisfies: R(k) > 2 k/^2

The Mean or Expected Value of a random variable X is the number E (X ) =

G ∈G(n,p) P(G^ ).X^ (G^ ).

The expected number of k-cycles in

G ∈ G(n, p), is E (X ) = ( 2 nk)k pk^.

Let k > 0 be an integer, and let p = p(n) be a function of n such that p ≥ (6k ln n)/n for large n. Then limn→∞P(α ≥ 2 nk ) = 0