The interplay between additive and symmetric large sets and their combinatorial applications

The study of symmetric structures is a new trend in Ramsey theory. Recently in [7], Di Nasso initiated a systematic study of symmetrization of classical Ramsey theoretical results, and proved a symmetric version of several Ramsey theoretic results. In this paper Di Nasso asked if his method could be adapted to find new non-linear Diophantine equations that are partition regular [7,Final remarks (4)]. By analyzing additive, multiplicative, and symmetric large sets, we construct new partition regular equations that give a first affirmative answer to this question. A special case of our result shows that if $P$ is a polynomial with no constant term then the equation $x+P(y-x)=z+w+zw$, where $y\mathrm{\ne }x$ is partition regular. Also we prove several new monochromatic patterns involving additive, multiplicative, and symmetric structures. Throughout our work, we use tools from the Algebra of the Stone-Čech Compactifications of discrete semigroups.