Blow-Up Lemma

The Regularity Lemma [16] is a powerful tool in Graph Theory and its applications. It basically says that every graph can be well approximated by the union of a constant number of random-looking bipartite graphs called regular pairs (see the definitions below). These bipartite graphs share many local properties with random bipartite graphs, e.g. most degrees are about the same, most pairs of vertices have about as many common neighbours as is expected in a random graph, and so on. These particular local properties imply other ones. Most importantly, they imply that every fixed bipartite graph H can be found as a subgraph in any large enough regular pair G (with a given positive density). This however is not an :exclusive property of regular pairs, or that of random graphs, since classical Extremal Graph Theory tells us that any given bipartite graph H is contained as a~subgraph in all large enough dense graphs (KSvs S6sTurs [8] and ErdSs-Stone [9]). The power of using the Regularity Lemma becomes apparent only when extended to much larger subgraphs H. Two examples are the theorem of Chvs [6] stating that all bounded degree graphs have linear Ramsey numbers, and the Alon-Yuster theorem [4] providing ahnost perfect coverings of a large G with copies of a small H.