Vector sum-intersection theorems

We introduce the following generalization of set intersection via characteristic vectors: for n,q,s,t≥1 a family F⊆{0,1,…,q}n vectors is said to be s-sum t-intersecting if any distinct x,y∈F there exist at least t coordinates, where entries x and y sum up s, i.e. |{i:xi+yi≥s}|≥t. The original corresponds case q=1,s=2. address analogs several variants classical results in this setting: Erdős–Ko–Rado theorem Bollobás on intersecting pairs.