The multicolor induced size-Ramsey number of long subdivisions

TL;DR

This paper establishes bounds on the multicolor induced size-Ramsey number for long subdivisions of graphs with bounded degree, showing it grows linearly with the number of vertices and depends on parameters like the number of colors and maximum degree.

Contribution

The authors derive new bounds for the multicolor induced size-Ramsey number of long subdivisions, extending previous results to more general graph classes and coloring scenarios.

Findings

Bound of ^{O(k \u2212   )} D^{9} (\u2212  D) n for general subdivisions.

Improved bound of O(k^{342} ( k)^9 D^{9} D) n when subdivision lengths are even.

Extended results to non-induced size-Ramsey numbers for long subdivisions.

Abstract

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.