Sets of nice recurrence are partition regular

A set of positive integers $R$ is called a set of nice recurrence if for any measure preserving system $(X,\mu ,T)$, for all measurable $A\subseteq X$, and each $\epsilon >0$, there exists $n\in R$ such that $\mu (A\cap T^{-n}A)⩾\mu (A{)}^2-\epsilon$. Answering a long-standing question of Bergelson, we show that sets of nice recurrence have the following Ramsey property: any finite colouring of a set of nice recurrence admits a monochromatic set of nice recurrence.