Given graphs $G$ and $H$, we say that $G$ is $H$-$good$ if the Ramsey number $R(G,H)$ equals the trivial lower bound $(|G| - 1)(\chi(H) - 1) + \sigma(H)$, where $\chi(H)$ denotes the usual chromatic number of $H$, and $\sigma(H)$ denotes the minimum size of a color class in a $\chi(H)$-coloring of $H$. Pokrovskiy and Sudakov [Ramsey goodness of paths. Journal of Combinatorial Theory, Series B, 122:384-390, 2017.] proved that $P_n$ is $H$-good whenever $n\geq 4|H|$. In this paper, given $\varepsilon>0$, we show that if $H$ satisfy a special unbalance condition, then $P_n$ is $H$-good whenever $n \geq (2 + \varepsilon)|H|$. More specifically, we show that if $m_1,\ldots, m_k$ are such that $\varepsilon\cdot m_i \geq 2m_{i-1}^2$ for $2\leq i\leq k$, and $n \geq (2 + \varepsilon)(m_1 + \cdots + m_k)$, then $P_n$ is $K_{m_1,\ldots,m_k}$-good.

翻译：给定图 $G$ 和 $H$，若拉姆齐数 $R(G,H)$ 等于平凡下界 $(|G| - 1)(\chi(H) - 1) + \sigma(H)$，则称 $G$ 是 $H$-$优$ 的，其中 $\chi(H)$ 表示 $H$ 的通常色数，$\sigma(H)$ 表示 $H$ 的 $\chi(H)$-染色中颜色类的最小尺寸。Pokrovskiy 与 Sudakov [Ramsey goodness of paths. Journal of Combinatorial Theory, Series B, 122:384-390, 2017.] 证明了当 $n\geq 4|H|$ 时，$P_n$ 是 $H$-优的。本文在给定 $\varepsilon>0$ 的条件下证明：若 $H$ 满足特定的不平衡条件，则当 $n \geq (2 + \varepsilon)|H|$ 时，$P_n$ 是 $H$-优的。具体而言，我们证明：若 $m_1,\ldots, m_k$ 满足 $\varepsilon\cdot m_i \geq 2m_{i-1}^2$（$2\leq i\leq k$）且 $n \geq (2 + \varepsilon)(m_1 + \cdots + m_k)$，则 $P_n$ 是 $K_{m_1,\ldots,m_k}$-优的。