Existence proofs in combinatorics using independence

TL;DR

This paper provides an accessible exposition on using the local Lovasz lemma to prove combinatorial results, illustrating the proof development process through problems and solutions for students with basic combinatorics knowledge.

Contribution

It demonstrates how to construct combinatorial existence proofs using the local Lovasz lemma, including the proof of the lemma itself, in an educational problem-based format.

Findings

Proofs of combinatorial results using the Lovasz lemma

Explanation of the independence notion and lemma proof

Problem-based presentation for educational purposes

Abstract

This note is purely expository and is in Russian. We show how to prove interesting combinatorial results using the local Lovasz lemma. The note is accessible for students having basic knowledge of combinatorics; the notion of independence is defined and the Lovasz lemma is stated and proved. Our exposition follows `Probabilistic methods' of N. Alon and J. Spencer. The main difference is that we show how the proof could have been invented. The material is presented as a sequence of problems, which is peculiar not only to Zen monasteries but also to advanced mathematical education; most problems are presented with hints or solutions.