The considered problem is combinatorial and it deals with arbitrary finite sets and colorings of their elements.
It shows that there exists a two-coloring of n elements such that n given sets on these elements have discrepancy at most Kn1/2.
The chief result, formulated in the language of linear forms, suddenly gives two corollaries, relevant to set theory and to classical Fourier analysis correspondingly.
The proof of the main theorem is based on the probabilistic method.
Paul Erdӧs is regarded as its founder.
Another interesting application of the main theorem is to the János Komlós Conjecture.
It gives a strong, albeit inconclusive, proof of the full suggestion.
It is important to prove that the main theorem is “best possible” up to the constant factor, i.e. to show, that the best asymptotic is obtained.
The main theorem is valid with moderate value of the constant K, K = 5.32.
November 2013
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