  One of the most basic results in graph theory states that every graph with at
least two vertices has two vertices with the same degree. Since there are
graphs without $3$ vertices of the same degree, it is natural to ask if for any
fixed $k$, every graph $G$ is ``close'' to a graph $G'$ with $k$ vertices of
the same degree. Our main result in this paper is that this is indeed the case.
Specifically, we show that for any positive integer $k$, there is a constant
$C=C(k)$, so that given any graph $G$, one can remove from $G$ at most $C$
vertices and thus obtain a new graph $G'$ that contains at least
$\min\{k,|G|-C\}$ vertices of the same degree.
  Our main tool is a multidimensional zero-sum theorem for integer sequences,
which we prove using an old geometric approach of Alon and Berman.
