Abstract: substructure will have a certain property. We will talk about two major results in Ramsey theory: van der Waerden’s Theorem and Hales-Jewett theorem. The focus will be on the proof structure of both. Since van der Waerden’s theorem guarantees k-term a monochromatic arithmetic progression for a large enough [n], then we will look to minimize monochromatic arithmetic progressions on various structures. One structure we will look at, is the coloring of [n]^2 . We will go through pseudocode for minimizing 3-term monochromatic arithmetic progressions in [n]^2 . After, we will see a picture of what the most optimal coloring may looks like. In theory, the code converges to the same coloring, each time.
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