The Multicolor Induced Size-Ramsey Number of Long Subdivisions

Abstract: For a positive integer $k$ and a graph $H$, the $k$-color induced size-Ramsey number \linebreak $\widehat{R}{\mathrm{ind}}(H, k)$ 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 $σ:E(H)\to \mathbb{N}$, the subdivision $H^σ$ is obtained by replacing each $e \in E(H)$ with a path of length $σ(e)$. 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~$H^σ$ be a subdivision of $H$ with $σ(e) > c \log_D n $ for every $e \in E(H)$, where~$n$ is the order of~$H^σ$. Then, $\hat{R}{\mathrm{ind}}(H^σ,k)=e^{O(k\log k)} D^{9}\log (D)\, n$. If each $σ(e)$ is even and larger than $c \log_D n$, this bound improves to $\hat{R}_{\mathrm{ind}}(H^σ,k)=O(k^{342} \log^{9} (k) D^{9} \log D )n$. We also find improved bounds for the non-induced size-Ramsey number of long subdivisions.