Representations of finite number of quadratic forms with same rank

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 $f_1,\dots ,f_k$ 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 $1\le m\le 8$,exceptforthecaseswhen$(m,n)=(3,5)$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.