The Multicolor Size-Ramsey Number of Bipartite Long Subdivisions

Abstract

For a positive integer $r$, the $r$-color size-Ramsey number~$\widehat{R}_r(H)$ of a graph $H$ is the minimum number of edges in a graph $G$ such that every $r$-edge coloring of $G$ contains a monochromatic copy of $H$. For a graph~$H$ and a function $\sigma:E(H)\to \mathbb{N}$, the \emph{subdivision} $H^\sigma$ is obtained by replacing every $e \in E(H)$ with a path of length $\sigma(e)$. In~\cite{javadi25:_induced_long} it is shown that for all integers $r,\, D\geq 2 $, there exists a constant $c=c(r, D)$ such that for every graph $ H $ with maximum degree $D$ if $H^{\sigma}$ is a subdivision of~$H$ in which $\sigma(e) > c \log n $ for every $e \in E(H)$, where $n=|V(H^\sigma)|$, then $ \widehat{R}_r(H^\sigma) = O\big(2^{34r} r^6 \log^5(r) D^5\log D\big)n. $ We improve upon this result in the case that~$H^{\sigma}$ is a bipartite graph and the number of colors~$r$ is large using a significantly different argument, obtaining the bound $ \widehat{R}_r(H^{\sigma}) \leq r^{400D \log D} \, n $.