Extremal results in random graphs

TL;DR

This paper surveys recent advances at the intersection of extremal combinatorics and random graph theory, highlighting the foundational contributions of Erdős and Turán and exploring new developments in the field.

Contribution

It provides a comprehensive overview of recent progress in extremal results within random graphs, emphasizing the historical context and recent research trends.

Findings

Summarizes key extremal results in random graphs

Highlights Erdős and Turán's foundational roles

Discusses recent research developments

Abstract

According to Paul Erd\H{o}s [Some notes on Tur\'an's mathematical work, J. Approx. Theory 29 (1980), page 4] it was Paul Tur\'an who "created the area of extremal problems in graph theory". However, without a doubt, Paul Erd\H{o}s popularized extremal combinatorics, by his many contributions to the field, his numerous questions and conjectures, and his influence on discrete mathematicians in Hungary and all over the world. In fact, most of the early contributions in this field can be traced back to Paul Erd\H{o}s, Paul Tur\'an, as well as their collaborators and students. Paul Erd\H{o}s also established the probabilistic method in discrete mathematics, and in collaboration with Alfr\'ed R\'enyi, he started the systematic study of random graphs. We shall survey recent developments at the interface of extremal combinatorics and random graph theory.