The multicolor induced size-Ramsey number of long subdivisions

For a positive integer $k$ and a graph $H$, the $k$-color induced size-Ramsey number $\hat{R}_{\mathrm{ind}}(H, k)$ is the minimum integer $m$ for which there exists a graph $G$ with $m$ edges such that for every $k$-edge coloring of $G$, the graph $G$ contains a monochromatic copy of $H$ as an induced subgraph. For a graph $H$ with the edge set $E(H)$ and a function $\sigma:E(H)\to \mathbb{N}$, the subdivision $H^\sigma$ is obtained by replacing each $e \in E(H)$ with a path of length $\sigma(e)$. We prove that for all integers $k,\, D\geq 2$, there exists a constant $c=c(k, D)$ such that the following holds. Let $H$ be any graph with maximum degree $D$ and let $H^{\sigma}$ be a subdivision of $H$ with $\sigma(e) > c \log_D n$ for every $e \in E(H)$, where $n$ is the order of $H^\sigma$. Then, $\hat{R}_{\mathrm{ind}}(H^\sigma,k)=e^{O(k\log k)} D^{9}(\log D)\, n$. If each $\sigma(e)$ is even and larger than $c \log_D n$, this bound improves to $\hat{R}_{\mathrm{ind}}(H^\sigma,k)=O(k^{342} (\log k)^9D^{9} \log D )n$. We also find improved bounds for the non-induced size-Ramsey number of long subdivisions.