MATH299E-The Ramsey’s Theorem States – Mathematics Assignment Help

Assignment Task

Task 

1. The restricted version of Ramsey’s theorem states that, given a target size n, there exists a size m such that no matter how one colors the edges of the graph Km with two colors, there will be a subgraph Kn for which all edges are the same color. We will now show an upper bound for m in terms of n.

(a) We define the function R(n) to be the smallest m such that Ramsey’s theorem applies, i.e. any coloring produces a monochromatic Kn. We also define a new notion R? (n, k) to be the smallest m such that any coloring of Km with colors RED and BLUE produces an all-RED Kn or an all-BLUE Kk subgraph. Note that R (n, n) = R(n) and that R (n, k) = R (k, n). Show the following inequality:

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