论文标题
Turán数字和反斑点的短周期，完整的3美元 - 明确图
Turán numbers and anti-Ramsey numbers for short cycles in complete $3$-partite graphs
论文作者
论文摘要
我们将$ 4 $ - 循环称为$ k_ {n_ {1}，n_ {2}，n_ {3}} $ suptratite，由$ c_ {4}^{\ text {multi}} $表示，如果它至少包含$ k_ {n _ {1}的每个部分， n_ {3}} $。 turán数字$ \ text {ex}（k_ {n_ {1}，n_ {2}，n_ {3}}，c_ {4}^{\ text {\ text {multi}}} $ \ text {ex}（k_ {n_ {1}，n_ {2}，n_ {3}}，\ {c_ {3}，c_ {4}^{\ text {\ text {multi}}}}} \} \}）$ k_ {n_ {1}，n_ {2}，n_ {3}} $，以至于$ g $包含no $ c_ {4}^{\ text {\ text {multi}} $ $ \ bigg（$分别，$ g $，$ g $都不包含$ c_ c_ {3} $ c_ {3} $ c_ nor $ c_ n} $ {4} $}如果所有四个边缘都有不同的颜色，我们将其称为$ c^{multi} _4 $彩虹。 Ant-ramsey数$ \ text {ar}（k_ {n_ {1}，n_ {2}，n_ {3}}，c_ {4}^{\ text {multi}}} $是$ k_ {n n n n n n n n n n n n n n n n n n n n n no n n n n n n n n n n n n n no {1}，n _ {2}，n _ {2） $ c_ {4}^{\ text {multi}} $。在本文中，我们确定$ \ text {ex}（k_ {n_ {1}，n_ {2}，n_ {3}}，c_ {4}^{\ text {multi}}} $ \ text {ar}（k_ {n_ {1}，n_ {2}，n_ {3}}，c_ {4}^{\ text {\ text {multi}}）= \ text {ex} c_ {4}^{\ text {multi}} \}）+1 = n_ {1} n_ {2}+n_ {3}+1，$ n_ {1} \ ge n_ {2} \ ge n_ {2}
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 $\text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})$ $\bigg($ respectively, $\text{ex}(K_{n_{1},n_{2},n_{3}},\{C_{3}, C_{4}^{\text{multi}}\})$ $\bigg)$ is the maximum number of edges in a graph $G\subseteq K_{n_{1},n_{2},n_{3}}$ such that $G$ contains no $C_{4}^{\text{multi}}$ $\bigg($ respectively, $G$ contains neither $C_{3}$ nor $C_{4}^{\text{multi}}$ $\bigg)$. We call a $C^{multi}_4$ 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 $n_{1}\ge n_{2}\ge n_{3}\ge 1.$
