The Green-Tao Theorem and a Relative Szemeredi Theorem
summary
Proves a relative Szemerédi result that removes the correlation condition from Green–Tao and replaces Gowers uniformity by a cut norm dense model, so fewer pseudorandom hypotheses are needed. Builds a sparse triangle/hypergraph counting lemma using repeated Cauchy–Schwarz and a new densification trick (convert co‑degrees to bounded edge weights), then transfers dense Roth/Szemerédi counts to the sparse setting. Applies to the Green–Tao strategy for primes by easing the verification of pseudorandom linear forms.
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