A note on non-empty cross-intersecting families

The families $\mathcal F_1\subseteq \binom{[n]}{k_1},\mathcal F_2\subseteq \binom{[n]}{k_2},\dots,\mathcal F_r\subseteq \binom{[n]}{k_r}$ are said to be cross-intersecting if $|F_i\cap F_j|\geq 1$ for any $1\leq i<j\leq r$ and $F_i\in \mathcal F_i$, $F_j\in\mathcal F_j$. Cross-intersecting families $\mathcal F_1,\mathcal F_2,\dots,\mathcal F_r$ are said to be non-empty if $\mathcal F_i\neq\emptyset$ for any $1\leq i\leq r$. This paper shows that if $\mathcal F_1\subseteq\binom{[n]}{k_1},\mathcal F_2\subseteq\binom{[n]}{k_2},\dots,\mathcal F_r\subseteq\binom{[n]}{k_r}$ are non-empty cross-intersecting families with $k_1\geq k_2\geq\cdots\geq k_r$ and $n\geq k_1+k_2$, then $\sum_{i=1}^{r}|\mathcal F_i|\leq\max\{\binom{n}{k_1}-\binom{n-k_r}{k_1}+\sum_{i=2}^{r}\binom{n-k_r}{k_i-k_r},\ \sum_{i=1}^{r}\binom{n-1}{k_i-1}\}$. This solves a problem posed by Shi, Frankl and Qian recently. The extremal families attaining the upper bounds are also characterized.