The blow-up lemma essentially says that regular pairs in Szemerédi's regularity lemma behave like complete bipartite graphs from the point of view of embedding bounded degree subgraphs.
In particular, given a graph 
of order 
, minimal vertex degree 
and maximal vertex
degree 
,
then there exists an 
such that the following holds. Let 
be an arbitrary positive integer, and replace the vertices
of 
with pairwise disjoint 
-sets 
, 
, ..., 
(blowing up). Now construct two graphs on the same vertex
set 
.
The graph 
is obtained by replacing all edges of 
with copies of the complete bipartite graph 
, and construct a sparser graph by replacing the edges
of 
with some 
-superregular pair. If a graph 
with 
is embeddable into 
, then it is already embeddable into 
(Komlós et al. 1998).
