r-Colour Size Ramsey Number

• The r-colour size Ramsey number is defined as the minimum number of edges in a host graph that guarantees every r-edge colouring produces a monochromatic copy of a given subgraph.
• Key results establish sharp bounds for paths, trees, cycles, and subdivisions, revealing dependencies on both the number of colours and the graph’s structural properties.
• Advanced proof techniques combine probabilistic constructions, recursive density extractions, and explicit expansion arguments to overcome traditional regularity lemma limitations.

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

$\widehat{R}_r(H) = \min\{ |E(G)| : \text{every %%%%13%%%%-edge-colouring of %%%%14%%%% contains a monochromatic copy of %%%%15%%%%} \}.$

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 (Beke et al., 20 Nov 2025).

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 $\widehat{R}_r(K_{1,m}) = r(m-1) + 1 = \Theta(r m)$ (DeBiasio, 8 Apr 2024).

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 $T$ (DeBiasio, 8 Apr 2024).

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 $n$ (Javadi et al., 2021). Earlier regularity-based results confirmed only the asymptotic linearity in $n$, with ill-understood dependence on $r$ (Javadi et al., 2017).

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 $n$ (Draganić et al., 2020). 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 $n$ (Letzter et al., 2021).

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 (Beke et al., 20 Nov 2025).
• 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 $n$ (Javadi et al., 2021).

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 (DeBiasio, 8 Apr 2024).

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 paper of monochromatic structure emergence in sparse random and explicit graphs, with connections spanning extremal graph theory, random combinatorics, and structural graph theory (Beke et al., 20 Nov 2025, DeBiasio, 8 Apr 2024, Javadi et al., 2021, Draganić et al., 2020, Letzter et al., 2021, Javadi et al., 2017).