Vjerojatnosne i funkcionalne varijante Szemeredijeve leme o regularnosti

This paper gives proofs of functional analytic and probabilistic variants of the Szemeredi regularity lemma and compares them with each other. The proof of the functional variant is based on the Hahn-Banach separation theorem, while the proof of the probabilistic variant is completely elementary and uses only basic probability theory. Throughout the thesis, Szemeredi’s theorem on arithmetic progressions provides the main source of the motivation. A special case of this theorem, Roth’s theorem, is proved using the probabilistic variant of regularity lemma.