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, …, qn of vectors is said to be s-sum t-intersecting if for any distinct x, y∈F there exist at least t coordinates, where the entries of x and y sum up to at least s, i. e. |i: xi+yi≥s|≥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 Erdős–Ko–Rado theorem and the theorem of Bollobás on intersecting set pairs.