  We introduce the following generalization of set intersection via
characteristic vectors: for $n,q,s, t \ge 1$ a family $\mathcal{F}\subseteq
\{0,1,\dots,q\}^n$ of vectors is said to be \emph{$s$-sum $t$-intersecting} if
for any distinct $\mathbf{x},\mathbf{y}\in \mathcal{F}$ there exist at least
$t$ coordinates, where the entries of $\mathbf{x}$ and $\mathbf{y}$ sum up to
at least $s$, i.e.\ $|\{i:x_i+y_i\ge s\}|\ge t$.
  The original set intersection corresponds to the case $q=1,s=2$.
  We address analogs of several variants of classical results in this setting:
the Erd\H{o}s--Ko--Rado theorem and the theorem of Bollob\'as on intersecting
set pairs.
