Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio

The regularity Lemma in additive combinatorics

The Szemerédi Regularity Lemma (SzRL) was introduced by Endré Szemerédi in his celebrated proof of the density version of Van der Waerden Theorem, namely, that a set of integers with positive density contains arbitrarily long arithmetic progressions. The SzRL has found applications in many areas of Mathematics, including of course Graph Theory and Combinatorics, but also in Number Theory, Analysis, Ergodic Theory and Computer Science. One of the consequences of the SzRL are the so-called `Counting Lemma' and `Removing Lemma', which roughly says that a sufficiently large graph G which contains not many copies of a fixed graph H can be made H-free by removing a small number of edges. Recently Ben Green gave an algebraic version of both, the SzRL and the Removal Lemma for groups. In this algebraic version the structural result fits into the algebraic structure in terms of subgroups. On the other hand, the Removal Lemma has its algebraic counterpart in the estimation of the number of solutions of equations in groups. The purpose of this Master Thesis is to give a detailed account on the SzRL and some of its applications, particularly to Additive Combinatorics. We particularly focuss on the consequences of the SzRL related to the Counting Lemma. By combining the version by Alon and Shapira of the directed version of the SzRL with the version of Simonovits for edge-colored graphs, we state and prove a Counting Lemma for arc-colored directed graphs. The methods used by Green heavily rely on Fourier Analysis, and as such, his results are applicable only to Abelian groups. By using our general version of the Counting Lemma we prove a generalization of Ben Green's Removal Lemma which is applicable to finite groups, non necessarily abelian