Monochromatic sums of squares

Abstract : For any integer let s(K) be the smallest integer such that in any colouring of the set of squares of the integers in K colours every large enough integer can be written as a sum of no more than s(K) squares, all of the same colour. A problem proposed by Sarkozy asks for optimal bounds for s(K) in terms of K. It is known by a result of Hegyvari and Hennecart that . In this article we show that . This improves on the bound , which is the best available upper bound for s(K).
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https://hal.archives-ouvertes.fr/hal-02060277
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Submitted on : Thursday, March 7, 2019 - 12:02:04 PM
Last modification on : Friday, March 8, 2019 - 1:28:48 AM

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Gyan Prakash, D. S. Ramana, Olivier Ramaré. Monochromatic sums of squares. Mathematische Zeitschrift, Springer, 2018, 289 (1-2), pp.51-69. ⟨10.1007/s00209-017-1943-7⟩. ⟨hal-02060277⟩

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