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In this expository note we present simple proofs of the lower bound of Ramseynumbers (Erd\"os theorem), and of the estimation of discrepancy. Neitherstatements nor proofs require any knowledge beyond high-school curriculum(except a minor detail). Thus they are accessible to non-specialists, inparticular, to students. Our exposition is simpler than the standard expositionbecause no probabilistic language is used. In order to prove the existence of a`good' object we prove that the number of `bad' objects is smaller than thenumber of all objects.

Simple proofs of estimations of Ramsey numbers and of discrepancy