  The set of degrees of maps $D(M,N)$, where $M,N$ are closed oriented
$n$-manifolds, always contains $0$ and the set of degrees of self-maps $D(M)$
always contains $0$ and $1$. Also, if $a,b\in D(M)$, then $ab\in D(M)$; a set
$A\subseteq\mathbb Z$ so that $ab\in A$ for each $a,b\in A$ is called
multiplicative. On the one hand, not every infinite set of integers (containing
$0$) is a mapping degree set [NWW] and, on the other hand, every finite set of
integers (containing $0$) is the mapping degree set of some $3$-manifolds
[CMV]. We show the following:
  (i) Not every multiplicative set $A$ containing $0,1$ is a self-mapping
degree set.
  (ii) For each $n\in\mathbb N$ and $k\geq3$, every $D(M,N)$ for $n$-manifolds
$M$ and $N$ is $D(P,Q)$ for some $(n+k)$-manifolds $P$ and $Q$.
  As a consequence of (ii) and [CMV], every finite set of integers (containing
$0$) is the mapping degree set of some $n$-manifolds for all $n\neq 1,2,4,5$.
