Sparse regularity and relative Szemerédi theorems

Abstract

We extend various fundamental combinatorial theorems and techniques from the dense setting to the sparse setting. First, we consider Szemerédi regularity lemma, a fundamental tool in extremal combinatorics. The regularity method, in its original form, is effective only for dense graphs. It has been a long standing problem to extend the regularity method to sparse graphs. We solve this problem by proving a so-called "counting lemma," thereby allowing us to apply the regularity method to relatively dense subgraphs of sparse pseudorandom graphs. Next, by extending these ideas to hypergraphs, we obtain a simplification and extension of the key technical ingredient in the proof of the celebrated Green-Tao theorem, which states that there are arbitrarily long arithmetic progressions in the primes. The key step, known as a relative Szemerédi theorem, says that any positive proportion subset of a pseudorandom set of integers contains long arithmetic progressions. We give a simple proof of a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Finally, we give a short simple proof of a multidimensional Szemerédi theorem in the primes, which states that any positive proportion subset of Pd (where P denotes the primes) contains constellations of any given shape. This has been conjectured by Tao and recently proved by Cook, Magyar, and Titichetrakun and independently by Tao and Ziegler.