Let \([n]{:}{=}\lbrace 1,2,\ldots ,n \rbrace \) , and M be a set of positive integers. Denote the family of all subsets of [n] with sizes in M by \(\left( {\begin{array}{c}\left[ n\right] \\ M\end{array}}\right) \) . The non-empty families \(\mathcal {A}\subseteq \left( {\begin{array}{c}\left[ n\right] \\ R\end{array}}\right) \) and \(\mathcal {B}\subseteq \left( {\begin{array}{c}\left[ n\right] \\ S\end{array}}\right) \) are said to be cross t-intersecting if \(|A\cap B|\ge t\) for all \(A\in \mathcal {A}\) and \(B\in \mathcal {B}\) . In this paper, we determine the maximum sum of sizes of non-empty cross t-intersecting families, and characterize the extremal families. Similar result for finite vector spaces is also proved.
