Order in the chaos with examples from graph theory

TL;DR

This paper explores the emergence of ordered substructures within random graphs, discussing key theorems and problems in graph theory such as Rédai's theorem, Ramsey theory, Turán's problem, and Hamiltonian cycles in De Bruijn graphs.

Contribution

It provides an overview of various fundamental results and examples illustrating order within chaos in graph theory structures.

Findings

Existence of ordered substructures in random graphs

Applications of Rédai's theorem and Ramsey theory

Insights into Hamiltonian cycles in De Bruijn graphs

Abstract

In randomly created structures (be they natural or artificial) very often there exist ordered substructures. In this Hungarian language scientific essay we will present some of such structures in graph theory. E.g. R\'edei's theorem, Ramsey theory, Tur\'an's problem, Hamiltonian cycles in De Bruijn graphs.