  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 $y\neq x$ 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-\v{C}ech
Compactifications of discrete semigroups.
