---
title: Gallai-Ramsey Number
url: https://www.emergentmind.com/topics/gallai-ramsey-number
type: topic
---

# Gallai-Ramsey Number

A Gallai-Ramsey number is an exact extremal invariant in multicolor Ramsey theory, quantifying the minimum order of a complete graph such that every edge-coloring, subject to a specified rainbow-avoidance constraint, must force a monochromatic copy of a given graph. For odd cycles, the Gallai-Ramsey number exhibits exceptionally clean behavior, contrasting sharply with the unpredictable growth of classical multicolor Ramsey numbers. The principal result is the closed formula $gr_k(K_3 : C_{2\ell+1}) = \ell \cdot 2^k + 1$, valid for all $k \geq 1$ and $\ell \geq 3$ [1808.09245]. The derivation of this formula rests on Gallai’s structure theorem for rainbow triangle-free colorings, inductive arguments, explicit extremal colorings, and a suite of tight path/cycle forcing lemmas.

## 1. Definition and Basic Properties

Given graphs $G$ and $H$, and a positive integer $k$, the $k$-color Gallai-Ramsey number $gr_k(G : H)$ is the least integer $N$ such that every $k$-edge-coloring of the complete graph $K_N$ contains either a rainbow copy of $G$ (all edges distinct colors) or a monochromatic copy of $H$. In the triangle-forced case, $H=C_{2\ell+1}$ and $G=K_3$, this measures the rainbow triangle or monochromatic odd cycle threshold [1808.09245]:

$$
gr_k(K_3 : C_{2\ell+1}) = \min\{N:\ \text{every $k$-coloring of } K_N\ \text{contains a rainbow } K_3 \text{ or a monochromatic } C_{2\ell+1}\}
$$

A $k$-coloring avoiding rainbow $K_3$ is called a Gallai coloring.

## 2. Main Result: Formula for Odd Cycles

For all $k \geq 1$ and $\ell \geq 3$,

$$
gr_k(K_3 : C_{2\ell+1}) = \ell \cdot 2^k + 1
$$

This formula is sharp—that is, there exist Gallai $k$-colorings on exactly $\ell \cdot 2^k$ vertices with neither a rainbow triangle nor a monochromatic $(2\ell +1)$-cycle. The result extends previous sharp values for $3 \leq \ell \leq 5$ and applies uniformly for all larger odd cycles [1808.09245]. The same formula (in the $k=3$ case) matches the conjectured value of the classical three-color Ramsey number for odd cycles ($R_3(C_{2\ell+1}) = 8\ell + 1$ when $k=3$).

## 3. Structure Theorem and Inductive Proof Outline

Gallai’s partition theorem underpins the entire analysis. It states that for any Gallai coloring of $K_n$, there exists a partition $V_1 \cup \dots \cup V_t$ such that between parts only two colors appear and every between-part edge is monochromatic. The reduced graph (one representative from each part) is thus a 2-colored complete graph.

The inductive argument for the upper bound proceeds as follows:

- **Base cases**: For $k=1$, the statement is trivial; one needs $2\ell+1 = \ell \cdot 2^1 + 1$ vertices. For $k=2$, the classical Ramsey number gives $gr_2(K_3 : C_{2\ell+1}) = 4\ell + 1 = \ell \cdot 4 + 1$.
- **Induction**: Assume the bound for $k-1$. Build sets $T_i$ for each color $i$ where every vertex in $T_i$ is color-$i$-complete to the rest. Each $|T_i|\leq \ell$ (as a larger $T_i$ would immediately yield a monochromatic $C_{2\ell+1}$). Remove all $T_i$ and analyze the Gallai partition of the remainder $G'$, which is structured by reduced graphs with only two colors and all parts of size at most $\ell$. By a series of pigeonhole, Dirac-type, and Erdős-Gallai lemmas, the presence of enough small parts plus classical two-colored extremal results ensures that $G'$ must contain a monochromatic $C_{2\ell+1}$, completing the argument [1808.09245].

## 4. Extremal Colorings and Sharpness

The matching lower bound is realized by an explicit construction:

- For $k=1,2$, use the identity coloring and the classical sharpness examples (two $K_{\ell}$ in red joined in blue, etc.).
- For $k \geq 3$, recursively take two disjoint copies of the extremal $(k-1)$-coloring on $\ell \cdot 2^{k-1}$ vertices, coloring all edges between in color $k$. This construction never creates a rainbow triangle, and the between-part edges are bipartite, hence cannot host an odd cycle. Any monochromatic odd cycle can only arise inside a part, which is precluded by induction [1808.09245].

## 5. Key Structural Lemmas

Several nontrivial classical results and structural sublemmas are critical in the analysis:

- **Dirac-type cycle lemma**: In a 2-colored complete graph on at least $2\ell+1$ vertices, if a color forms a connected spanning subgraph with minimum degree at least $n/2$, a monochromatic $C_{2\ell+1}$ appears.
- **Erdős-Gallai path lemma (2-color version)**: If every vertex $v$ in a 2-colored graph satisfies $d_{red}(v) + d_{blue}(v) \geq a+b-3$, then the graph contains either a red path on $a$ vertices or a blue path on $b$ vertices.
- **Path splicing and pigeonhole arguments**: In reduced graphs with two large parts or a large part and the rest small, path and cycle embedding becomes possible by augmenting long monochromatic paths to close cycles [1808.09245].

## 6. Specific Examples and Small Cases

For $k=2$, $\ell=3$, $gr_2(K_3 : C_7) = 13$. The sharp example is two red $K_6$’s joined in blue. Each part contains no red $C_7$; the blue subgraph is bipartite and thus contains no odd cycles. For $k \geq 3$, recursive blowup constructions extend this extremal pattern.

| $k$ | $\ell$ | $gr_k(K_3: C_{2\ell+1})$ |
|-----|--------|--------------------------|
|  1  |   3    | 7                        |
|  2  |   3    | 13                       |
|  3  |   3    | 25                       |
|  2  |   4    | 17                       |
|  3  |   4    | 33                       |

## 7. Extensions, Open Questions, and Context

The odd-cycle Gallai-Ramsey result represents a special instance of a broader dichotomy:

- For non-bipartite $H$, $gr_k(K_3: H)$ grows exponentially in $k$.
- For bipartite $H$ not a star, $gr_k(K_3: H)$ grows linearly in $k$.
- For stars, $gr_k(K_3: H)$ is constant in $k$ [1905.11794].

For even cycles $C_{2\ell}$, only linear-in-$k$ upper and lower bounds are known; the exact value is open. Replacing $K_3$ with larger forbidden rainbow graphs dramatically increases structural complexity, and the general $gr_k(G: H)$ problem remains wide open beyond isolated exact instances. Removing the rainbow triangle constraint yields the classical multicolor Ramsey numbers, but even for cycles, these problems are notoriously difficult [1808.09245].

The combination of structural rainbow-avoidance (Gallai partitions), classical extremal graph theory, and careful recursive coloring analysis enables these exact results for odd cycles, resolving a longstanding conjecture and establishing a canonical exponential law:

$$
gr_k(K_3 : C_{2\ell+1}) = \ell \cdot 2^k + 1
$$

for all $k,\, \ell \geq 1$ with $\ell \geq 3$ [1808.09245].

Source: https://www.emergentmind.com/topics/gallai-ramsey-number