Tur\'an numbers and anti-Ramsey numbers for short cycles in complete $3$-partite graphs

We call a $4$-cycle in $K_{n_1,n_2,n_3}$ multipartite, denoted by $C_4^{\text{multi}}$, if it contains at least one vertex in each part of $K_{n_1,n_2,n_3}$. The Tur'an number $$\text{ex}$(K_{n_{1},n_{2},n_{3}}, C_{4}^{$\text{multi}$})$ $($ respectively, $\text{ex}(K_{n_1,n_2,n_3},\{C_3,C_4^{\text{multi}}\})$ $)$ is the maximum number of edges in a graph $G\subseteq K_{n_1,n_2,n_3}$ such that $G$ contains no $C_4^{\text{multi}}$ $($ respectively, $G$ contains neither $C_3$ nor $C_4^{\text{multi}}$ $)$. We call a $C_4^{multi}$ rainbow if all four edges of it have different colors. The ant-Ramsey number $\text{ar}(K_{n_1,n_2,n_3},C_4^{\text{multi}})$ is the maximum number of colors in an edge-colored of $K_{n_1,n_2,n_3}$ with no rainbow $C_4^{\text{multi}}$. In this paper, we determine that $\text{ex}(K_{n_1,n_2,n_3},C_4^{\text{multi}})=n_1{n}_2+2n_3$ and $$\text{ar}$(K_{n_{1},n_{2},n_{3}}, C_{4}^{$\text{multi}$})=$\text{ex}$(K_{n_{1},n_{2},n_{3}}, {C_{3}, C_{4}^{$\text{multi}$}})+1=n_{1}n_{2}+n_{3}+1,$where$n_{1}\ge n_{2}\ge n_{3}\ge 1.$