---
title: r-Colour Size Ramsey Number
url: https://www.emergentmind.com/topics/r-colour-size-ramsey-number
type: topic
---

# r-Colour Size Ramsey Number

The $r$-colour size Ramsey number of a graph $H$, denoted $\widehat{R}_r(H)$, is the minimum number of edges in a host graph $G$ such that every edge-colouring of $G$ with $r$ colours yields a monochromatic copy of $H$. This parameter bridges classical Ramsey theory and extremal combinatorics, considering both the structure of the target graph $H$ and the sparsity constraints of $G$. The quantity captures fundamental thresholds for monochromatic substructure emergence in sparse random or pseudorandom settings, with pronounced differences across classes including paths, cycles, trees, subdivisions, and hypergraphs.

## 1. Definitions and Basic Properties

Let $H$ be a fixed (finite) graph and $r\geq 2$ an integer. The $r$-colour size Ramsey number $\widehat{R}_r(H)$ is defined as
\[
\widehat{R}_r(H) = \min\{ |E(G)| : \text{every $r$-edge-colouring of $G$ contains a monochromatic copy of $H$} \}.
\]
This notion generalizes the classical vertex Ramsey number $R_r(H)$ by focusing on edge-minimality rather than vertex proliferation. For vertex-disjoint graphs $G_1,\ldots,G_r$, the (multi)colour size Ramsey number $\widehat{R}(G_1,\ldots,G_r)$ denotes the least $m$ such that every $r$-colouring of the edges of $G$ (with $|E(G)|=m$) yields a monochromatic copy of some $G_i$ in its assigned colour.

Key structural graph parameters relevant in size Ramsey problems include the **vertex cover number** $\tau(H)$ and, for trees, the bipartition-based parameter $\beta(T) = n_1\Delta_1 + n_2\Delta_2$, with $V(T)=V_1\cup V_2$, $n_i=|V_i|$, $\Delta_i = \max_{v\in V_i} d_T(v)$.

## 2. Paths, Trees, and Connected Graphs: Core Results

The $r$-colour size Ramsey number of paths, trees, and general connected graphs is governed by sharp lower and upper bounds, with key distinctions arising from global structural constraints.

### Paths

Recent work of Beke, Li, and Sahasrabudhe shows that for $r\geq 2$ and $k\geq 100 \log r$,
\[
\widehat{R}_r(P_k) = \Theta\bigl((r^2 \log r)\,k\bigr),
\]
closing the previous logarithmic gap and demonstrating the necessity of the $\log r$ factor for all $r$. The proof employs a refined recursive density extraction process (via “balls-and-bins” heavy tail estimates), yielding improved lower bounds matching the random-graph construction-based upper bound [2511.16656].

### General Connected Graphs

For all non-star connected graphs $H$ with $m$ edges and $r \geq 6$ (sufficiently large $m$), DeBiasio establishes
\[
\widehat{R}_r(H) \geq \frac{r^2}{64} m = \Omega(r^2 m),
\]
with the only exception being the star $K_{1,m}$, which satisfies $\widehat{R}_r(K_{1,m}) = r(m-1) + 1 = \Theta(r m)$ [2404.05856].

### Trees

For every non-star tree $T$, with bipartition parameter $\beta(T)$, the lower bound
\[
\widehat{R}_r(T) \geq \frac{r^2}{2048}\beta(T)
\]
is complemented by a matching upper bound for “$\alpha$-full” trees, i.e., those for which at least one part in the bipartition contains a vertex of degree linear in the opposite part’s size:
\[
\widehat{R}_r(T) = \Theta(r^2 \beta(T)).
\]
These results subsume prior work (Beck, Dellamonica, Krivelevich) and clarify the tight dependence on both $r$ and the combinatorial structure of $T$ [2404.05856].

## 3. Cycles, Parity, and Size Ramsey Thresholds

The size Ramsey behaviour for cycles displays a deep parity dichotomy, reflected in the dependence of $\widehat{R}_r(C_n)$ on $n$ and $r$.

For all $r\geq 2$,
- **Even $n$:** There exist absolute constants $c_1, c_2 > 0$ such that
  \[
  c_1 r^2 n \leq \widehat{R}_r(C_n) \leq c_2 r^{120} (\log^2 r) n.
  \]
- **Odd $n$:** There exist $c_1, c_2 > 0$ such that
  \[
  c_1 2^{r} n \leq \widehat{R}_r(C_n) \leq c_2 2^{16 r^{2} + 2 \log r} n.
  \]

The lower bounds derive from the corresponding (tight) path Ramsey bounds, while the upper bounds follow from direct “quasi-random” host graph constructions avoiding the Szemerédi regularity lemma, with explicit combinatorial concentration arguments yielding the correct order for even $n$ and nearly optimal order for odd $n$ [2106.16023]. Earlier regularity-based results confirmed only the asymptotic linearity in $n$, with ill-understood dependence on $r$ [1701.07348].

## 4. Size Ramsey Numbers for Subdivisions and Hypergraphs

For $q$-subdivisions $H^q$ of a bounded-degree graph $H$ (each edge of $H$ is replaced by a path of length $q$), Draganić, Krivelevich, and Nenadov prove
\[
\widehat{R}_r(H^q) \leq C n^{1 + 1/q}
\]
for some $C = C(\Delta, q, r)$ and host graphs of size $n$ [2004.14139]. The proof uses threshold-random graphs, sparse regularity, and robust expansion/embedding arguments, optimizing the exponent (and removing previous $\log n$ factors). The lower bound is matched in exponent by first-moment considerations.

In the hypergraph regime, Letzter, Pokrovskiy, and Yepremyan show that tight powers and long subdivisions of bounded-degree graphs and hypergraphs preserve linear size Ramsey numbers, provided the relevant subdivision parameter is polylogarithmic in $n$ [2103.01942].

## 5. Proof Techniques and Key Lemmas

Modern approaches to multicolour size Ramsey bounds leverage a combination of probabilistic and combinatorial frameworks:

- **Sparse random graph host construction:** For given $k$ and $r$, the Erdős–Rényi $G(n, p)$ or explicit random bipartite/regular models with tuned $p$ and $n$ yield typical hosts of minimal edge count, with high-probability expansion or anti-bipartite-hole properties.
- **Colour-splitting and random partitioning:** Balanced iterative colour partitioning and edge assignment, often exploiting affine-plane or tree expansion structures, avoid unwanted monochromatic subgraphs.
- **Recursive extraction and density increments:** Bootstrapping via recursive extraction of maximal forbidden-subgraph-free edge sets (“balls-and-bins” argument) and density improvement at each scale; star and Vizing-type colourings eliminate high-degree obstructions [2511.16656].
- **Tree and cycle embedding lemmas:** Embedding trees in expanders (Friedman–Pippenger theorem), and using strong local expansion/expander concentration to guarantee long paths or cycles in pseudorandom host graphs.
- **Avoidance of the regularity lemma:** Elementary Chernoff and expansion techniques are now preferred, as they produce explicit constants and polynomial or nearly optimal bounds in $r$ and $n$ [2106.16023].

## 6. Exceptional Cases, Open Problems, and Implications

The star $K_{1,m}$ is a unique exception to the general $\Omega(r^2 m)$ threshold, satisfying only $\widehat{R}_r(K_{1,m}) = r(m-1)+1$. No other graph family exhibits similar subquadratic in $r$ behaviour [2404.05856].

Remaining open problems include:
- Sharpening constant factors and dependence on $r$ (e.g., determining if the log-factor in $\widehat{R}_r(P_k)$ is intrinsic or can be universally eliminated).
- Understanding bounds for powers of hypergraph trees and subdivisions for general, possibly unbounded, degree hosts.
- Pinning down exact structural thresholds for bounded-degree graphs where superlinear $n$ growth occurs (no such example is currently known except for hypergraphs with large powers or subdivision parameter).
- Resolving the precise behaviour of size Ramsey numbers for odd-length cycles in the multicolour regime, particularly the necessity of exponential in $r$ growth.

These results underpin the study of monochromatic structure emergence in sparse random and explicit graphs, with connections spanning extremal graph theory, random combinatorics, and structural graph theory [2511.16656, 2404.05856, 2106.16023, 2004.14139, 2103.01942, 1701.07348].

Source: https://www.emergentmind.com/topics/r-colour-size-ramsey-number