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.
Categories: math.CO