Gallai-Ramsey Number

• Gallai-Ramsey number is an extremal invariant in multicolor Ramsey theory that defines the minimum complete graph order forcing a rainbow triangle or monochromatic odd cycle.
• The closed formula grₖ(K₃ : C₂ₗ₊₁) = ℓ·2ᵏ + 1 illustrates its exponential growth for k ≥ 1 and ℓ ≥ 3, providing sharp bounds.
• The derivation leverages Gallai’s partition theorem, inductive arguments, and extremal coloring constructions to avoid rainbow triangles.

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 $\ell \geq 3$ (Wang et al., 2018). 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 (Wang et al., 2018):

$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 (Wang et al., 2018). 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 (Wang et al., 2018).

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 (Wang et al., 2018).

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 (Wang et al., 2018).

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 $k$ (Magnant et al., 2019).

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 (Wang et al., 2018).

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 $\ell \geq 3$ (Wang et al., 2018).