A strong Gurarii space is a Banach space containing isometric copies of all finite-dimensional spaces which is additionally homogeneous with respect to finite-dimensional subspaces. The latter means that every linear isometry between its finite-dimensional subspaces extends to a bijective isometry of the entire space. It is well-known (already noticed by Gurarii) that no separable Banach space can be a strong Gurarii space. On the other hand, there exist strong Gurarii spaces of density at least the continuum. This leads to a natural set-theoretic question whether consistently one can have a strong Gurarii space of a smaller density. We show that a strong Gurarii space of density aleph actually exists in ZFC .

This is a joint work with Antonio Aviles.