Infinite families of manifolds of positive $k^{\mathrm{t}\mathrm{h}}$-intermediate Ricci curvature with $k$ small

Positive $k^{\mathrm{t}\mathrm{h}}$-intermediate Ricci curvature on a Riemannian $n$-manifold, to be denoted by ${\mathrm{R}\mathrm{i}\mathrm{c}}_k>0$, is a condition that interpolates between positive sectional and positive Ricci curvature (when $k =1$and$k=n-1$ respectively). In this work, we produce many examples of manifolds of ${\mathrm{R}\mathrm{i}\mathrm{c}}_k>0$ with $k$ small by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension $n\ge 7$ congruent to $3\mathrm{m}\mathrm{o}\mathrm{d}4$ supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of ${\mathrm{R}\mathrm{i}\mathrm{c}}_k>0$ for some $k<n\mathrm{/}2$. We also prove that each dimension $n\ge 4$ congruent to $0$ or $1\mathrm{m}\mathrm{o}\mathrm{d}4$ supports closed manifolds which carry metrics of ${\mathrm{R}\mathrm{i}\mathrm{c}}_k>0$ with $k\le n\mathrm{/}2$, but do not admit metrics of positive sectional curvature.