A SPARSE REGULAR APPROXIMATION LEMMA

We introduce a new variant of Szemeredi's regularity lemma which we call the sparse regular approximation lemma (SRAL). The input to this lemma is a graph G of edge density p and parameters epsilon, delta, where we think of delta as a constant. The goal is to construct an &regular partition of G while having the freedom to add/remove up to delta vertical bar IE(G)vertical bar edges. As we show here, this weaker variant of the regularity lemma already suffices for proving the graph removal lemma and the hypergraph regularity lemma, which are two of the main applications of the (standard) regularity lemma. This of course raises the following question: can one obtain quantitative bounds for SRAL that are significantly better than those associated with the regularity lemma? Our first result answers the above question affirmatively by proving an upper bound for SRAL given by a tower of height O(log 1/p). This allows us to reprove Fox's upper bound for the graph removal lemma. Our second result is a matching lower bound for SRAL showing that a tower of height Omega(log 1/p) is unavoidable. We in fact prove a more general multicolored lower bound which is essential for proving lower bounds for the hypergraph regularity lemma.