Multicolor Ramsey numbers on stars versus pat

Abstract: For given simple graphs $H_1,H_2,\dots,H_c$, the multicolor Ramsey number $R(H_1,H_2,\dots,H_c)$ is defined as the smallest positive integer $n$ such that for an arbitrary edge-decomposition ${G_i}^c_{i=1}$ of the complete graph $K_n$, at least one $G_i$ has a subgraph isomorphic to $H_i$. Let $m,n_1,n_2,\dots,n_c$ be positive integers and $\Sigma=\sum_{i=1}^{c}(n_i-1)$. Some bounds and exact values of $R(K_{1,n_1},\dots,K_{1,n_c},P_m)$ have been obtained in literature. Wang (Graphs Combin., 2020) conjectured that if $\Sigma\not\equiv 0\pmod{m-1}$ and $\Sigma+1\ge (m-3)^2$, then $R(K_{1,n_1},\ldots, K_{1,n_c}, P_m)=\Sigma+m-1.$ In this note, we give a new lower bound and some exact values of $R(K_{1,n_1},\dots,K_{1,n_c},P_m)$ when $m\leq\Sigma$, $\Sigma\equiv k\pmod{m-1}$, and $2\leq k \leq m-2$. These results partially confirm Wang's conjecture.