Simple proofs of estimations of Ramsey numbers and of discrepancy

TL;DR

This paper provides accessible, simple proofs for lower bounds of Ramsey numbers and discrepancy estimates, avoiding advanced probabilistic methods to make the results understandable to non-specialists.

Contribution

It introduces straightforward, high-school level proofs for key combinatorial bounds, simplifying existing complex probabilistic approaches.

Findings

Proved the lower bounds of Ramsey numbers using elementary methods.

Estimated discrepancy without probabilistic language.

Made proofs accessible to students and non-specialists.

Abstract

In this expository note we present simple proofs of the lower bound of Ramsey numbers (Erd\"os theorem), and of the estimation of discrepancy. Neither statements nor proofs require any knowledge beyond high-school curriculum (except a minor detail). Thus they are accessible to non-specialists, in particular, to students. Our exposition is simpler than the standard exposition because 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 the number of all objects.