The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Large independent sets from local considerations

The following natural problem was raised independently by Erd\H{o}s-Hajnal and Linial-Rabinovich in the late 80's. How large must the independence number $\alpha(G)$ of a graph $G$ be whose every $m$ vertices contain an independent set of size $r$? In this paper we discuss new methods to attack this problem. The first new approach, based on bounding Ramsey numbers of certain graphs, allows us to improve previously best lower bounds due to Linial-Rabinovich, Erd\H{o}s-Hajnal and Alon-Sudakov. As an example, we prove that any $n$-vertex graph $G$ having an independent set of size $3$ among every $7$ vertices has $\alpha(G) \ge \Omega(n^{5/12})$. This confirms a conjecture of Erd\H{o}s and Hajnal that $\alpha(G)$ should be at least $n^{1/3+\varepsilon}$ and brings the exponent half-way to the best possible value of $1/2$. Our second approach deals with upper bounds. It relies on a reduction of the original question to the following natural extremal problem. What is the minimum possible value of the $2$-density of a graph on $m$ vertices having no independent set of size $r$? This allows us to improve previous upper bounds due to Linial-Rabinovich, Krivelevich and Kostochka-Jancey. As part of our arguments we link the problem of Erd\H{o}s-Hajnal and Linial-Rabinovich and our new extremal $2$-density problem to a number of other well-studied questions. This leads to many interesting directions for future research.Comment: 26 pages in the main body, 7 figures, 3 appendice

A survey of χ\chi-boundedness

If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made a number of challenging conjectures about this in the early 1980's, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now