Arithmetic Ramsey theory over the primes

Abstract: We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$, where the $a_i$ and $b$ are fixed coefficients, and $h$ is an arbitrary integer polynomial of degree $d$. Provided there are at least $(1+o(1))d^2$ variables, we establish necessary and sufficient criteria for this equation to have a monochromatic non-constant solution with respect to any finite colouring of the prime numbers. We similarly characterise when such equations admit solutions over any set of primes with positive relative upper density. In both cases, we obtain counting results which provide asymptotically sharp lower bounds for the number of monochromatic or dense solutions in primes. Our main new ingredient is a uniform lower bound on the cardinality of a prime polynomial Bohr set.