Matchings Meeting Quotas and Their Impact on the Blow-Up Lemma

A bipartite graph G = (U,V;E) is called $\epsilon$-regular if the edge density of every sufficiently large induced subgraph differs from the edge density of G by no more than $\epsilon$. If, in addition, the degree of each vertex in G is between $(d-\epsilon)n$ and $(d+\epsilon)n$, where d is the edge density of G and |U|=|V|=n, then G is called super $(d,\epsilon)$-regular. In [Combinatorica, 19 (1999), pp. 437--452] it was shown that if $S \subset U$ and $T \subset V$ are subsets of vertices in a super-regular bipartite graph G = (U,V;E), and if a perfect matching M of G is chosen randomly, then the number of edges of M that go between the sets S and T is roughly |S||T|/n. In this paper, we derandomize this result using the Erdos--Selfridge method of conditional probabilities. As an application, we give an alternative constructive proof of the blow-up lemma of $\komlos$, $\sarkozy$, and $\szemeredi$ (see [Combinatorica, 17 (1997), pp. 109--123] and [Random Structures Algorithms, 12 (1998), pp. 297--312]).