Corners theorem

Corners theorem In arithmetic combinatorics, the corners theorem states that for every {displaystyle varepsilon >0} , for large enough {displaystyle N} , any set of at least {displaystyle varepsilon N^{2}} points in the {displaystyle Ntimes N} grid {displaystyle {1,ldots ,N}^{2}} contains a corner, i.e., a triple of points of the form {displaystyle {(x,y),(x+h,y),(x,y+h)}} with {displaystyle hneq 0} . 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] Contents 1 Statement 2 Proof overview 3 Quantitative bounds 4 Multidimensional extension 4.1 Multidimensional Szemerédi's Theorem 5 References 6 External links Statement Define a corner to be a subet of {displaystyle mathbb {Z} ^{2}} of the form {displaystyle {(x,y),(x+h,y),(x,y+h)}} , where {displaystyle x,y,hin mathbb {Z} } and {displaystyle hneq 0} . For every {displaystyle varepsilon >0} , there exists a positive integer {displaystyle N(varepsilon )} such that for any {displaystyle Ngeq N(varepsilon )} , any subset {displaystyle Asubseteq {1,ldots ,N}^{2}} with size at least {displaystyle varepsilon N^{2}} contains a corner.

The condition {displaystyle hneq 0} can be relaxed to {displaystyle h>0} by showing that if {displaystyle A} is dense, then it has some dense subset that is centrally symmetric.

Proof overview What follows is a sketch of Solymosi's argument.

Suppose {displaystyle Asubset {1,ldots ,N}^{2}} is corner-free. Construct an auxiliary tripartite graph {displaystyle G} with parts {displaystyle X={x_{1},ldots ,x_{N}}} , {displaystyle Y={y_{1},ldots ,y_{N}}} , and {displaystyle Z={z_{1},ldots ,z_{2N}}} , where {displaystyle x_{i}} corresponds to the line {displaystyle x=i} , {displaystyle y_{j}} corresponds to the line {displaystyle y=j} , and {displaystyle z_{k}} corresponds to the line {displaystyle x+y=k} . Connect two vertices if the intersection of their corresponding lines lies in {displaystyle A} .

Note that a triangle in {displaystyle G} corresponds to a corner in {displaystyle A} , except in the trivial case where the lines corresponding to the vertices of the triangle concur at a point in {displaystyle A} . It follows that every edge of {displaystyle G} is in exactly one triangle, so by the triangle removal lemma, {displaystyle G} has {displaystyle o(|V(G)|^{2})} edges, so {displaystyle |A|=o(N^{2})} , as desired.

Quantitative bounds Let {displaystyle r_{angle }(N)} be the size of the largest subset of {displaystyle [N]^{2}} which contains no corner. The best known bounds are {displaystyle {frac {N^{2}}{2^{(c_{1}+o(1)){sqrt {log _{2}N}}}}}leq r_{angle }(N)leq {frac {N^{2}}{(log log N)^{c_{2}}}},} where {displaystyle c_{1}approx 1.822} and {displaystyle c_{2}approx 0.0137} . 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 {displaystyle mathbb {Z} ^{d}} is a set of points of the form {displaystyle {a}cup {a+he_{i}:1leq ileq d}} , where {displaystyle e_{1},ldots ,e_{d}} is the standard basis of {displaystyle mathbb {R} ^{d}} , and {displaystyle hneq 0} . 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 {displaystyle Ssubseteq mathbb {Z} ^{d}} , and for every {displaystyle varepsilon >0} , there exists a positive integer {displaystyle N(S,varepsilon )} such that for any {displaystyle Ngeq N(S,varepsilon )} , any subset {displaystyle Asubseteq {1,ldots ,N}^{d}} with size at least {displaystyle varepsilon N^{d}} contains a subset of the form {displaystyle acdot S+h} . This theorem follows from the multidimensional corners theorem by a simple projection argument.[6] In particular, Roth's theorem follows directly from the ordinary corners theorem.

References ^ Ajtai, Miklós; Szemerédi, Endre (1974). "Sets of lattice points that form no squares". Stud. Sci. Math. Hungar. 9: 9–11. MR 0369299.. ^ 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. ^ Green, Ben (2021). "Lower Bounds for Corner-Free Sets". arXiv:0710.3032 [math.CO]. ^ 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. ^ 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. ^ Jump up to: a b 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. ^ 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. External links Proof of the corners theorem on polymath. Categories: 1974 introductions1974 in mathematicsRamsey theoryAdditive combinatoricsTheorems in combinatorics