A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma

Details A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma

2006-02-02

arxiv.org/abs/math/0602037v2

Mathematics

Combinatorics

39 pages, to appear, J. d'Analyse Mathematique. Proof of the relative independence property simplified; many suggestions of th

Scientific paper

We introduce a correspondence principle (analogous to the Furstenberg correspondence principle) that allows one to extract an infinite random graph or hypergraph from a sequence of increasingly large deterministic graphs or hypergraphs. As an application we present a new (infinitary) proof of the hypergraph removal lemma of Nagle-Schacht-R\"odl-Skokan and Gowers, which does not require the hypergraph regularity lemma and requires significantly less computation. This in turn gives new proofs of several corollaries of the hypergraph removal lemma, such as Szemer\'edi's theorem on arithmetic progressions.

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