  Let $m, n$ be positive integers with $m\le n$. Let $\kappa(m,n)$ be the
largest integer $k$ such that for any (positive definite and integral)
quadratic forms $f_1,\ldots,f_k$ of rank $m$, there exists a quadratic form of
rank $n$ that represents $f_i$ for any $i$ with $1\le i \le k$. In this
article, we determine the number $\kappa(m,n)$ for any integer $m$ with $1\le
m\le 8$, except for the cases when $(m,n)=(3,5)$ and $(4,6)$. In the
exceptional cases, it will be proved that $1\le \kappa(3,5), \ \kappa(4,6)\le
2$. We also discuss some related topics.
