Turán numbers and anti-Ramsey numbers for short cycles in complete \(3\)-partite graphs (Q6162143)

Summary: 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 \(\text{ex}(K_{n_{1}, n_{2}, n_{3}}, C_4^{\text{multi}})\) (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 \(C^{\text{multi}}_4\) (respectively, \(G\) contains neither \(C_3\) nor \(C^{\text{multi}}_4\)). We call an edge-colored \(C^{\text{multi}}_4\) rainbow if all four edges of it have different colors. The anti-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 \(K_{n_{1}, n_{2}, n_{3}}\) with no rainbow \(C^{\text{multi}}_4\). 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 \(n_1\geqslant n_2 \geqslant n_3 \geqslant 1\).