Random graphs and the probabilistic method

The main source used in the preparation of this study is Diestel's (2017) "Graph Theory" book. This thesis mainly consists of studying the eleventh chapter of the aforementioned book to understand and explain the subject of "random graphs and the probabilistic method". But of course, no part of the mentioned book has been quoted exactly, a study has been put forward with our own words and our own sentences; and on the one hand, some very difficult or technical proofs were skipped, on the other hand, some parts of the book that were left to the reader were explained and the subject was presented more understandably. In addition to these, the articles listed in the references were also consulted. To summarize in outline: In the first chapter, basic definitions and theorems of graph theory, which are the prerequisites for understanding the main subject, are covered by using the first chapter of the aforementioned book. In the first section of the second chapter, the concept of a random graph and related definitions and results are given. In the second section, the probabilistic method pioneered by Erdös in his article published in 1959 is introduced, and the related Erdös (1959) theorem is proved. Random graphs have results that are random even in their expressions, and these are results that contain the phrase "almost all". In the third section of the second chapter, the properties of "almost all graphs" are studied and the Erdös and Renyi (1963) theorem is proved. In the fourth and last section, the threshold functions and second moments are introduced and the threshold function of the graph property containing a given graph is determined (Erdös and Renyi 1960, Bollobas 1981). In the third chapter of the thesis, before the conclusions and recommendation, a brief literature review on the subject of the thesis is presented by using the notes section of the eleventh chapter of the mentioned book.