Vector sum-intersection theorems (2305.01328v1)

Abstract: 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.