Ramsey Goodness of paths and unbalanced graphs

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.