Guaranteed by Szemerédi's Regularity Lemma, a technique originated by Łuczak is to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component. So determining the Ramsey number of connected matchings is crucial in determining the Ramsey number of cycles. Let $k,l,m$ be integers and $r\left(k,l,m\right)$ be the minimum integer $N$ such that for any red‐blue‐green coloring of $K_{N,N}$, there is a red matching of size at least $k$ in a component, or a blue matching of size at least $l$ in a component, or a green matching of size at least $m$ in a component. Bucić, Letzter, and Sudakov determined the exact value of $r\left(k,l,l\right)$ and led to the asymptotic value of 3‐colored bipartite Ramsey number of cycles (symmetric case). In this paper, we determine the exact value of $r\left(k,l,m\right)$ completely. This answers a question of Bucić, Letzter, and Sudakov. The crucial part of the proof is the construction we give in Section 4. Applying the technique of Łuczak, we obtain the asymptotic value of 3‐colored bipartite Ramsey number of cycles for all asymmetric cases.

中文翻译：

---

三色不对称二分拉姆西连接匹配和循环数

由Szemerédi的正则引理保证，Łuczak发起的一项技术是减少显示单色循环的存在以显示组件中单色匹配的存在的问题。因此，确定连接匹配的Ramsey数对于确定循环的Ramsey数至关重要。让$ķ，升，米$ 是整数和 $\mathrm{[R}（ķ，升，米）$ 是最小整数 $ñ$ 这样对于任何红色-蓝色-绿色 ${ķ}_{ñ，ñ}$，至少有一个红色大小匹配 $ķ$ 在一个组件中，或大小至少为蓝色的匹配项 $升$ 在组件中，或者绿色匹配的大小至少 $米$在一个组件中。Bucić，Letzter和Sudakov确定了$\mathrm{[R}（ķ，升，升）$并导致3色二分Ramsey循环数的渐近值（对称情况）。在本文中，我们确定$\mathrm{[R}（ķ，升，米）$完全。这回答了Bucić，Lettzter和Sudakov的问题。证明的关键部分是我们在第4节中给出的构造。应用Łuczak技术，我们获得了所有不对称情况下三色二分拉姆西循环数的渐近值。