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    • Theses

    • 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.

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