On intersection forms of definite 4-manifolds bounded by a rational homology 3-sphere

We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by $Y$. To this end, we make use of constraints on definite forms bounded by $Y$ induced from Donaldson's diagonalization theorem, and correction term invariants due to Fr\o yshov, and Ozsv\'ath and Szab\'o. In particular, we prove that all spherical 3-manifolds satisfy such finiteness property.