Monochromatic integers adding to polynomials of prime variables
A theorem of \textit{A.~Khalfalah} and \textit{E.~Szemerédi} [Comb.\ Probab.\ Comput. 15, No. 1--2, 213--227 (2006; Zbl 1093.11017)] states: Let \(\psi\) be a (non-constant) polynomial with integral coefficients and positive leading coefficient. Suppose that \(\psi(1)\psi(0)\) is even. Then for any \(m\)-colouring of all positive integers, there exist monochromatic distinct \(x,y\) such that \(x+y=\psi(z)\) for an integer \(z\). Also, \textit{B.~Green} [Ann.\ Math. (2) 161, No. 3, 1609--1636 (2005; Zbl 1160.11307)] has proved that for a set of primes of positive relative density, there are infinitely many triples \((x,y,z)\) of primes with \(x+y=2z\). In the paper under review the authors use related methods to prove the following result: Let \(m,b_0,W_0\) be positive integers satisfying \(1 \leq b_0 \leq W_0\) and \((b_0,W_0) = 1\). Let \(\psi(x)\) be a (non-constant) polynomial with integral coefficients and positive leading coefficient satisfying that \(\psi(1)\) or \(\psi(0)\) is even if \(2 \mid W_0\), or \(\psi(b_0 - 1)\) is even if \(2 \nmid W_0\). Suppose that all positive integers are coloured with \(m\) colours. Then there exist distinct monochromatic \(x,y\) such that \(x+y=\psi(z)\) for a prime \(z \equiv b_0 \bmod W\). A related results is shown for triples of primes: Suppose that for each prime \(p\), there exists an integer \(c_p\) such that \(\psi(c_p)\) is even, and neither \(W_0c_p +b_0\) nor \(\frac{1}{2}\psi(c_p)\) is divisible by \(p\). Then for any \(m\)-colouring of all primes, there exist distinct monochromatic primes \(x,y\) such that \(x+y=\psi(z)\) for a prime \(z \equiv b_0 \bmod W\).
scientific article; zbMATH DE number 5990943
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