Abstract
The basic problem of (combinatorial) discrepancy theory is how to color a set with two colors as uniformly as possible with respect to a given family of subsets. The aim is to achieve that each of the two colors meets each subset under consideration in approximately the same number of elements. From the finite Ramsey theorem (cf. Corollary 7.2) we know already that if the set of all 2-subsets of n is 2-colored, and the family of all ℓ-subsets for some \(\ell< \frac{1} {2}\log n\) is considered, the situation is as bad as possible: for any 2-coloring we will find a monochromatic ℓ-set. As ℓ gets larger one can color more uniformly though one still has the preponderance phenomenon.
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Prömel, H.J. (2013). A Quasi Ramsey Theorem. In: Ramsey Theory for Discrete Structures. Springer, Cham. https://doi.org/10.1007/978-3-319-01315-2_10
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