Turán 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 Turán number of $ C_{4}^{\text{multi}})$ and $\{C_{3}, C_{4}^{\text{multi}}\})$ is determined in the paper as well as the anti-Ramsey number $\text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})$. We prove 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 $n_{1}\ge n_{2}\ge n_{3}\ge 1.$