Different Contexts in Ramsey Theory
April 22, 2005
Pizza and drinks will be served
Three main theorems exist within the field of Ramsey Theory. Ramsey's theorem itself, which says that when you have a significantly large graph, you can always find a large complete subgraph or a large independent subgraph. Van der Waerden's theorem says that if you 2-color a large initial segment of the natural numbers, then you can always find a large, monochromatic arithmetic progression. Finally, Hindman's theorem says that if you 2-color a large initial segment of the natural numbers, there is a large subset of it such that all the non repeating sums from this subset are the same color. We look at each of these theorems in several different contexts. For example, what happens if we use an infinite number of colors? What happens on the real line?
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