The Multicolor Induced Size-Ramsey Number of Long Subdivisions

TL;DR

This work proves explicit upper bounds for the multicolor induced size-Ramsey numbers of long subdivisions of bounded-degree graphs, showing that if each edge of the subdivided graph is stretched beyond a threshold $\sigma(e)>c\log_{D} n$, then $\widehat{R}_{\mathrm{ind}}(H^{\sigma},k)$ scales linearly with the order $n$ up to a factor depending on $k$ and $D$, namely $e^{O(k\log k)} D^{9}\log D\, n$, with a substantially sharper bound in the even-length subdivision case. The authors develop a regularity-free framework built on random linear hypergraphs with sparsity guarantees, gadget graphs $F$ that enforce short cycles, and a carefully controlled embedding scheme that pre-emptively reserves resources for critical vertices. The method hinges on constructing an expander host graph via auxiliary graphs $G$ and $G'$, extending induced-good embeddings through a pre-emptive greedy process, and a three-phase embedding strategy (growing trees, connecting them, and rolling back) to realize $H^{\sigma}$ as an induced subgraph of a gadget-augmented host. The results advance the understanding of induced Ramsey phenomena for subdivisions and provide concrete, scalable bounds in the induced setting, with companion work addressing related non-induced and bipartite cases.

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.

Figures (4)

• Figure 1: Critical Vertices. The neighbors $v_1$, $v_2$ of $v$ are not induced-available since $h(vv_1)\setminus \{v\}$ and $h(vv_2)\setminus \{v\}$ intersect $V(J)$. Also, $v_3$ and $v_d$ are not induced-available because $h(vv_3)\setminus \{v\}$ intersects $\mathop{\mathrm{cl}}\nolimits(J)$ and $h(vv_d)\setminus \{v\}$ intersects $\mathop{\mathrm{cl}}\nolimits_2(J)$.
• Figure 2: Extension of an induced-good embedding.
• Figure 3: Scheme of the embedded graph.
• Figure 4: Subgraphs $J_i$ and $K_i$ and the forbidden set $F_i$. Red (bold) edges are edges of $K_i\setminus J_i$.

Theorems & Definitions (7)

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• proof : Proof of Lemma \ref{['lem:subgraph']}
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