Notes on Extremal Graph Theory

Lecture Notes Spring 2006 M 690 I : Extremal Graph Theory

Extremal Results in Random Graphs

According to Paul Erdős [Some notes on Turán’s mathematical work, J. Approx. Theory 29 (1980), page 4] it was Paul Turán who “created the area of extremal problems in graph theory”. However, without a doubt, Paul Erdős popularized extremal combinatorics, by his many contributions to the field, his numerous questions and conjectures, and his influence on discrete mathematicians in Hungary and al...

Regularity lemmas in a Banach space setting

Szemerédi’s regularity lemma is a fundamental tool in extremal graph theory, theoretical computer science and combinatorial number theory. Lovász and Szegedy [7] gave a Hilbert space interpretation of the lemma and an interpretation in terms of compactness of the space of graph limits. In this paper we prove several compactness results in a Banach space setting, generalising results of Lovász a...

Some new results in extremal graph theory

In recent years several classical results in extremal graph theory have been improved in a uniform way and their proofs have been simpli…ed and streamlined. These results include a new Erd1⁄2os-Stone-Bollobás theorem, several stability theorems, several saturation results and bounds for the number of graphs with large forbidden subgraphs. Another recent trend is the expansion of spectral extrem...

Topics in Extremal Combinatorics - Notes

3 Lecture 3 8 3.1 Krivelevich’s proof that r(3, k) ≥ ck2/ log k . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Ajtai-Komlós-Szemerédi’s r(3, k) < ck2/ log k . . . . . . . . . . . . . . . . . . . . . . 9 3.3 AKS Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 AKS formal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...