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

A bipartite graph G = (U, V ;E) is called ǫ-regular if the edge density of every sufficiently large induced subgraph differs from the edge density of G by no more than ǫ. If, in addition, the degree of each vertex in G is between (d− ǫ)n and (d+ ǫ)n, where d is the edge density of G and |U | = |V | = n, then G is called super (d, ǫ)-regular. In [Combinatorica, 19 (1999), pp. 437– 452] it was shown that if S ⊂ U and T ⊂ 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 Erdős–Selfridge method of conditional probabilities. As an application, we give an alternative constructive proof of the blow-up lemma of Komlós, Sárközy, and Szemerédi (see [Combinatorica, 17 (1997), pp. 109–123] and [Random Structures Algorithms, 12 (1998), pp. 297–312]).