Corners theorem

In arithmetic combinatorics, the corners theorem states that for every , for large enough , any set of at least points in the grid contains a corner, i.e., a triple of points of the form with . It was first proved by Miklós Ajtai and Endre Szemerédi in 1974 using Szemerédi's theorem.[1] In 2003, József Solymosi gave a short proof using the triangle removal lemma.[2]

Statement

Define a corner to be a subset of of the form , where and . For every , there exists a positive integer such that for any , any subset with size at least contains a corner.

The condition can be relaxed to by showing that if is dense, then it has some dense subset that is centrally symmetric.

Proof overview

What follows is a sketch of Solymosi's argument.

Suppose is corner-free. Construct an auxiliary tripartite graph with parts , , and , where corresponds to the line , corresponds to the line , and corresponds to the line . Connect two vertices if the intersection of their corresponding lines lies in .

Note that a triangle in corresponds to a corner in , except in the trivial case where the lines corresponding to the vertices of the triangle concur at a point in . It follows that every edge of is in exactly one triangle, so by the triangle removal lemma, has edges, so , as desired.

Quantitative bounds

Let be the size of the largest subset of which contains no corner. The best known bounds are

where and . The lower bound is due to Green,[3] building on the work of Linial and Shraibman.[4] The upper bound is due to Shkredov.[5]

Multidimensional extension

A corner in is a set of points of the form , where is the standard basis of , and . The natural extension of the corners theorem to this setting can be shown using the hypergraph removal lemma, in the spirit of Solymosi's proof. The hypergraph removal lemma was shown independently by Gowers[6] and Nagle, Rödl, Schacht and Skokan.[7]

Multidimensional Szemerédi's Theorem

The multidimensional Szemerédi theorem states that for any fixed finite subset , and for every , there exists a positive integer such that for any , any subset with size at least contains a subset of the form . This theorem follows from the multidimensional corners theorem by a simple projection argument.[6] In particular, Roth's theorem on arithmetic progressions follows directly from the ordinary corners theorem.

References

  1. Ajtai, Miklós; Szemerédi, Endre (1974). "Sets of lattice points that form no squares". Stud. Sci. Math. Hungar. 9: 9–11. MR 0369299..
  2. Solymosi, József (2003). "Note on a generalization of Roth's theorem". In Aronov, Boris; Basu, Saugata; Pach, János; et al. (eds.). Discrete and computational geometry. Algorithms and Combinatorics. Vol. 25. Berlin: Springer-Verlag. pp. 825–827. doi:10.1007/978-3-642-55566-4_39. ISBN 3-540-00371-1. MR 2038505.
  3. Green, Ben (2021). "Lower Bounds for Corner-Free Sets". arXiv:0710.3032 [math.CO].
  4. Linial, Nati; Shraibman, Adi (2021). "Larger Corner-Free Sets from Better NOF Exactly-N Protocols". Discrete Analysis. 2021. arXiv:2102.00421. doi:10.19086/da.28933. S2CID 231740736.
  5. Shkredov, I.D. (2006). "On a Generalization of Szemerédi's Theorem". Proceedings of the London Mathematical Society. 93 (3): 723–760. arXiv:math/0503639. doi:10.1017/S0024611506015991. S2CID 55252774.
  6. Gowers, Timothy (2007). "Hypergraph regularity and the multidimensional Szemerédi theorem". Annals of Mathematics. 166 (3): 897–946. arXiv:0710.3032. doi:10.4007/annals.2007.166.897. MR 2373376. S2CID 56118006.
  7. Rodl, V.; Nagle, B.; Skokan, J.; Schacht, M.; Kohayakawa, Y. (2005-05-26). "From The Cover: The hypergraph regularity method and its applications". Proceedings of the National Academy of Sciences. 102 (23): 8109–8113. Bibcode:2005PNAS..102.8109R. doi:10.1073/pnas.0502771102. ISSN 0027-8424. PMC 1149431. PMID 15919821.
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