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Bootstrapping partition regularity of linear systems. (English) Zbl 1448.05204

Summary: Suppose that \(A\) is a \(k \times d\) matrix of integers and write \(\operatorname{Re}_A : \mathbb{N} \to \mathbb{N} \cup \{\infty\}\) for the function taking \(r\) to the largest \(N\) such that there is an \(r\)-colouring \(\mathcal{C}\) of \([N]\) with \(\bigcup_{C \in \mathcal{C}} C^d \cap \ker A = \emptyset\). We show that if \(\operatorname{Re}_A(r) < \infty\) for all \(r \in \mathbb{N}\) then \(\operatorname{Re}_A(r) \leqslant \exp (\exp (r^{O_A(1)}))\) for all \(r \geq 2\). When the kernel of \(A\) consists only of Brauer configurations – that is, vectors of the form \((y, x, x + y, \ldots, x + (d - 2)y)\) – the above statement has been proved by J. Chapman and S. Prendiville [Bull. Lond. Math. Soc. 52, No. 2, 316–334 (2020; Zbl 1442.05232)] with good bounds on the \(O_A (1)\) term.

MSC:

05D10 Ramsey theory
11B30 Arithmetic combinatorics; higher degree uniformity

Citations:

Zbl 1442.05232
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References:

[1] Abbott, H. L. and Moser, L., Sum-free sets of integers, Acta Arith.11 (1966), 393-396. · Zbl 0151.03701
[2] Ahmed, T. and Schaal, D. J., On generalized Schur numbers, Exp. Math.25(2) (2016), 213-218. · Zbl 1403.11018
[3] Beutelspacher, A. and Brestovansky, W., Generalized Schur numbers, in Combinatorial theory (Schloss Rauischholzhausen, 1982) (eds Jungnickel, D. and Vedder, K.), pp. 30-38, , Volume 969 (Springer, Berlin, 1982). · Zbl 0498.05002
[4] Bourgain, J., On triples in arithmetic progression, Geom. Funct. Anal.9(5) (1999), 968-984. · Zbl 0959.11004
[5] Chapman, J., Partition regularity and multiplicatively syndetic sets, preprints (arXiv.org/abs/1902.01149, 2019).
[6] Chapman, J. and Prendiville, S., On the Ramsey number of the Brauer configuration, preprint (arXiv.org/abs/1904.07567v1, 2019). · Zbl 1442.05232
[7] Chow, S., Lindqvist, S. and Prendiville, S., Rado’s criterion over squares and higher powers, preprint (arXiv.org/abs/1806.05002, 2018).
[8] Cwalina, K. and Schoen, T., Tight bounds on additive Ramsey-type numbers, J. Lond. Math. Soc.96(3) (2017), 601-620. · Zbl 1431.11024
[9] Deuber, W., Partitionen und lineare Gleichungssysteme, Math. Z.133 (1973), 109-123. · Zbl 0254.05011
[10] Frankl, P., Graham, R. L. and Rödl, V., Quantitative theorems for regular systems of equations, J. Combin. Theory Ser. A47(2) (1988), 246-261. · Zbl 0654.05002
[11] Gasarch, W., Moriarty, R. and Tumma, N., New upper and lower bounds on the Rado numbers, preprint (arXiv.org/abs/1206.4885, 2012).
[12] Gowers, W. T., A new proof of Szemerédi’s theorem for arithmetic progressions of length four, Geom. Funct. Anal.8(3) (1998), 529-551. · Zbl 0907.11005
[13] Gowers, W. T., A new proof of Szemerédi’s theorem, Geom. Funct. Anal.11(3) (2001), 465-588. · Zbl 1028.11005
[14] Gowers, W. T. and Wolf, J., The true complexity of a system of linear equations, Proc. Lond. Math. Soc. (3)100(1) (2010), 155-176. · Zbl 1243.11010
[15] Graham, R. L. and Rothschild, B. L., Ramsey’s theorem for n-parameter sets, Trans. Amer. Math. Soc.159 (1971), 257-292. · Zbl 0233.05003
[16] Graham, R. L., Rothschild, B. L. and Spencer, J. H., Ramsey theory, 2nd edn, (John Wiley & Sons, New York, 1990). · Zbl 0705.05061
[17] Green, B. J., Finite field models in additive combinatorics, in Surveys in combinatorics 2005, pp. 1-27, , Volume 327 (Cambridge University Press, Cambridge, 2005). · Zbl 1155.11306
[18] Green, B. J. and Lindqvist, S., Monochromatic solutions to x + y = z^2, Can. J. Math.71(3) (2019), 579-605. · Zbl 1439.11076
[19] Green, B. J. and Sanders, T., Monochromatic sums and products, Discrete Anal. (5) (arXiv:1510.08733, 2016). doi: doi:10.19086/da.613. · Zbl 1400.11024
[20] Green, B. J. and Tao, T. C., An arithmetic regularity lemma, an associated counting lemma, and applications, in An irregular mind, pp. 261-334, , Volume 21 (János Bolyai Mathematical Society, Budapest, 2010). · Zbl 1222.11015
[21] Green, B. J. and Tao, T. C., Linear equations in primes, Ann. Math. (2)171(3) (2010), 1753-1850. · Zbl 1242.11071
[22] Gunderson, D. S., On Deuber’s partition theorem for (m, p, c)-sets, Ars Combin.63 (2002), 15-31. · Zbl 1072.05505
[23] Gupta, S., Thulasi Rangan, J. and Tripathi, A., The two-colour Rado number for the equation ax + by = (a + b)z, Ann. Comb.19(2) (2015), 269-291. · Zbl 1317.05116
[24] Hatami, H. and Lovett, S., Higher-order Fourier analysis of \(\mathbb{F}^n_p\) and the complexity of systems of linear forms, Geom. Funct. Anal.21(6) (2011), 1331-1357. · Zbl 1291.11026
[25] Hindman, N. and Leader, I. B., Nonconstant monochromatic solutions to systems of linear equations, in Topics in discrete mathematics, pp. 145-154, , Volume 26 (Springer, Berlin, 2006). · Zbl 1117.05106
[26] Landman, B. M. and Robertson, A., Ramsey theory on the integers, 2nd edn, , Volume 73 (American Mathematical Society, Providence, RI, 2014). · Zbl 1307.05218
[27] Lê, T. H., Partition regularity and the primes, C. R. Math.350(9) (2012), 439-441. · Zbl 1275.11023
[28] Li, H. and Pan, H., A Schur-type addition theorem for primes, J. Number Theory132(1) (2012), 117-126. · Zbl 1276.11016
[29] Manners, F., Good bounds in certain systems of true complexity one, Discrete Anal. (21) (arXiv:1705.06801, 2018). doi: doi:10.19086/da.6814. · Zbl 1429.11024
[30] Prendiville, S., Quantitative bounds in the polynomial Szemerédi theorem: the homogeneous case, Discrete Anal.5 (2017), 1-34. · Zbl 1404.11114
[31] Rado, R., Studien zur Kombinatorik, Math. Z.36 (1933), 424-480. · Zbl 0006.14603
[32] Rado, R., Some partition theorems, in Combinatorial theory and its applications, III, Proc. Colloq., Balatonfüred, 1969, pp. 929-936 (János Bolyai Mathematical Society, 1970). · Zbl 0216.29804
[33] Robertson, A. and Myers, K., Some two color, four variable Rado numbers, Adv. Appl. Math.41(2) (2008), 214-226. · Zbl 1178.05095
[34] Sanders, J. H., A generalization of Schur’s theorem. PhD thesis, Yale University, 1968. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:6908427
[35] Saracino, D., The 2-color Rado number of x_1 + x_2 + · · · + x_n = y_1 + y_2 + · · · + y_k, Ars Combin.129 (2016), 315-321. · Zbl 1413.05372
[36] Schur, I., Über die Kongruenz x^m + y^m ≡ z^m (mod.p), Jahresber. Dtsch. Math.-Ver.25 (1916), 114-117. · JFM 46.0193.02
[37] Shkredov, I. D., Fourier analysis in combinatorial number theory, Russ. Math. Surv.65(3) (2010), 513-567. · Zbl 1270.11010
[38] Tao, T. C., Higher order Fourier analysis, , Volume 142 (American Mathematical Society, Providence, RI, 2012). · Zbl 1277.11010
[39] Tao, T. C. and Vu, V. H., Additive combinatorics, , Volume 105 (Cambridge University Press, Cambridge, 2006). · Zbl 1127.11002
[40] Van Der Waerden, B. L., Beweis einer Baudetschen Vermutung, Nieuw Arch. Wiskd., II. Ser.15 (1927), 212-216. · JFM 53.0073.12
[41] Walker, A., Gowers norms control Diophantine inequalities, preprint (arXiv.org/abs/1703. 00885), 2018. · Zbl 1478.11052
[42] Wolf, J., Finite field models in arithmetic combinatorics – ten years on, Finite Fields Appl.32 (2015), 233-274. · Zbl 1378.11021
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