... journal of combinatorics 10 (2003), #R39 62.1 An Invariant of Szemer´edi’s Regularity LemmaLet G =(U ∪ V, E) be a bipartite graph. For an integer t, we define a t-equitable partition V (G) as a ... journal of combinatorics 10 (2003), #R39 10[6] J. Skokan and L. Thoma, Bipartite subgraphs and quasi-randomness, accepted toGraphs and Combinatorics.[7] E. Szemer´edi, Regular partitions of graphs, ... state, up front, that we alwaysassume m is a sufficiently large integer.Our proof of Theorem 1.5 uses a well-known invariant formulation of Szemer´edi’sRegularity Lemma. We now present that formulation.2As...