Abstract

Szemerédi's regularity lemma is one of the most celebrated results in modern graph theory. However, in its original setting it is only helpful for studying large dense graphs, that is, graphs with n vertices and Θ(n2) edges. The main reason for this is that the underlying concept of ε-regularity is not meaningful when dealing with sparse graphs, since for large enough n every graph with o(n2) edges is ε-regular. In 1997 Kohayakawa and Rödl independently introduced a modified definition of ε-regularity which is also useful for sparse graphs, and used it to prove an analogue of Szemerédi's regularity lemma for sparse graphs. However, some of the key tools for the application of the regularity lemma in the dense setting, the so-called embedding lemmas or, in their stronger forms, counting lemmas, are not known to be true in the sparse setting. In fact, counterexamples show that these lemmas do not always hold. However, Kohayakawa, Luczak, and Rödl formulated a probabilistic embedding lemma that, if true, would solve several long-standing open problems in random graph theory. In this survey we give an introduction to Szemerédi's regularity lemma and its generalisation to the sparse setting, describe embedding lemmas and their applications, and discuss recent progress towards a proof of the probabilistic embedding lemma. In particular, we present various properties of ε-regular graphs in the sparse setting. We also show how to use these results to prove a weak version of the conjectured probabilistic embedding lemma.