Asymmetrical Ramsey Bounds for Hypercubes
نویسنده
چکیده
An n-cube is the set of 2 points in n-dimensional Euclidean space and is often denoted by {0, 1}n. It is known that n-cube as a Cartesian product of pair of points is Ramsey, but no asymmetrical results are known. In this work we are concerned with bounds on the necessary dimension n of Euclidean space such that every r-coloring gives us monochromatic hypercube of some color and given dimension respective to that color. We will give an upper for a special family of cases where all but one hypercubes we look for are degenerated. We will also show lower bounds for two cases with just 1and 2-dimensional cubes involved. The results obtained are closely related to the famous problem of chromatic number of Euclidean space.
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