A complete classification of the rational realization of gap-two products of Eilenberg--MacLane spaces as classifying spaces

In this paper, we study the rational realization of the gap-two product $K(\text{Q},n)\times K(\text{Q},n+2)$ as the classifying space $\text{B}(X)$ for a simply-connected $\pi$-finite space $X$ with rational homology of finite type. We prove that $K(\text{Q},n)\times K(\text{Q},n+2)$ can be realized as $\text{B}(X)$ up to rational homotopy equivalence if and only if $n=3,4$ or $4m+2$ for $m\ge 1$. Moreover, we determine all their minimal Sullivan models up to isomorphism. The rational homotopy type of such $X$ is unique for $n=3$ or $4m+2$ for $m\ge 1$; there exist infinitely many rational homotopy types of such $X$ for $n=4$.