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Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory pp 173–180Cite as

Triangle Removal and Szemerédi Regularity

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2239))

Abstract

In this chapter, we give nonstandard proofs of two of the more prominent results in extremal graph theory, namely the Triangle Removal Lemma and the Szemerédi Regularity Lemma.

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Notes

  1. 1.

    Given a binary relation R on a set X, we write R −1 for the binary relation on X given by (x, y) ∈ R −1 if and only if (y, x) ∈ R.

  2. 2.

    Here, for \(f\in L^2(\mathcal {L}_{V\times V})\), \(\mathbb {E}[f|\mathcal {L}_V\otimes \mathcal {L}_V]\) denotes the conditional expectation of f onto the subspace \(L^2(\mathcal {L}_V\otimes \mathcal {L}_V)\).

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  4. E. Szemerédi, On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27, 199–245 (1975)

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Author information

Authors and Affiliations

  1. Department of Mathematics, Universita di Pisa, Pisa, Italy

    Mauro Di Nasso

  2. Department of Mathematics, University of California, Irvine, Irvine, CA, USA

    Isaac Goldbring

  3. School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand

    Martino Lupini

Authors
  1. Mauro Di Nasso
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  2. Isaac Goldbring
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  3. Martino Lupini
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Nasso, M.D., Goldbring, I., Lupini, M. (2019). Triangle Removal and Szemerédi Regularity. In: Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory. Lecture Notes in Mathematics, vol 2239. Springer, Cham. https://doi.org/10.1007/978-3-030-17956-4_16

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