On indefinite $k$-universal integral quadratic forms over number fields

An integral quadratic lattice is called indefinite $k$-universal if it represents all integral quadratic lattices of rank $k$ for a given positive integer $k$. For $k\geq 3$, we prove that the indefinite $k$-universal property satisfies the local-global principle over number fields. For $k=2$, we show that a number field $F$ admits an integral quadratic lattice which is locally $2$-universal but not indefinite 2-universal if and only if the class number of $F$ is even. Moreover, there are only finitely many classes of such lattices over $F$. For $k=1$, we prove that $F$ admits a classic integral lattice which is locally classic $1$-universal but not classic indefinite $1$-universal if and only if $F$ has a quadratic unramified extension where all dyadic primes of $F$ split completely. In this case, there are infinitely many classes of such lattices over $F$. All quadratic fields with this property are determined.