Online Ramsey Number Overview

• Online Ramsey number is defined as the minimal rounds in a sequential coloring game where Builder forces a monochromatic copy of a target graph under an adversarial Painter response.
• The methodology employs Builder strategies that exploit forced Painter moves through path extensions, cycle shortenings, and case-specific gadgets to hasten graph completions.
• Key results include asymptotic bounds and exact thresholds for various graph pairs, highlighting superlinear savings over brute-force constructions and novel insights into game variants.

The online Ramsey number is a central invariant of a two-player sequential coloring game on a host graph (frequently the infinite clique $K_\mathbb{N}$), played between Builder and Painter. In each round, Builder selects a previously unexposed edge; Painter immediately assigns a color (typically red or blue, but sometimes more) to that edge. Builder's objective is to force the appearance of a monochromatic copy of a target graph $H$ (or, in the general two-graph setting, to force a red copy of $G_1$ or a blue copy of $G_2$) as rapidly as possible, while Painter tries to delay this outcome. The online Ramsey number, denoted $\tilde{r}(G_1,G_2)$, is the minimal number of rounds required for Builder to guarantee victory under optimal adversarial painter response.

1. Formal Definition and Game Variants

The standard online Ramsey game is played on an infinite vertex set, where every round consists of:

• Builder selecting an unused edge,
• Painter coloring the edge red or blue.

The online Ramsey number is then

$$\tilde{r}(G_1, G_2) = \min \left\{ N : \text{Builder has a strategy to force a red } G_1 \text{ or a blue } G_2 \text{ within } N \text{ moves} \right\}$$

with $G_1,G_2$ fixed target graphs (Zhang et al., 2023).

Variants include restricted online Ramsey numbers, played on a finite host graph of $N$ vertices (with all edges initially uncolored), and ordered online Ramsey games, where the vertex set is equipped with a linear order and monochromatic subgraphs must preserve this order (Dvořák, 2020, Gonzalez et al., 2018, Heath et al., 2024).

A further extension is the online Ramsey turnaround game, where in each round Builder may forbid one color on the exposed edge, Painter picks from the remaining colors, and the game's objective is reversed: Painter aims to force a monochromatic copy of the target $H$ as quickly as possible (Almási et al., 8 Dec 2025).

2. Relationship to Classical Ramsey and Size Ramsey Numbers

The online Ramsey number interpolates between the vertex Ramsey number $r(G_1, G_2)$ and the size Ramsey number $H$0: $H$1 (Zhang et al., 2023). All Builder strategies in the online Ramsey number setting can be viewed as sequential constructions of host graphs with the Ramsey property; optimal strategies often exploit forced responses to accelerate the appearance of $H$2, yielding superlinear savings over brute-force constructions (Gonzalez et al., 2018).

3. Key Results and Asymptotic Estimates

Off-diagonal Path-Path Online Ramsey Numbers

Recent research has determined:

• For $H$3 and $H$4,

$H$5

with exact lower and upper bounds proven (Mond et al., 2023, Adamska et al., 21 Apr 2025, Bednarska-Bzdȩga, 2023). Prior conjectures positing a $H$6 factor are disproven for $H$7.

• For $H$8 vs $H$9,

$G_1$0

(Zhang et al., 2023, Bednarska-Bzdȩga, 2023).

Cycle-Cycle and Path-Cycle Online Ramsey Numbers

• For cycles $G_1$1 (even $G_1$2) vs long cycles $G_1$3, the tight bound is: $G_1$4 (Adamski et al., 2023).
• For the claw $G_1$5 vs cycles: $G_1$6 (Zhi et al., 9 Jan 2026).
• For $G_1$7 vs paths, the threshold is: $G_1$8 (Adamski et al., 2022). Exact value for $G_1$9 is determined as $G_2$0 (Litka, 2023).

Diagonal Clique Online Ramsey Numbers

Sharp lower bounds have been obtained: $G_2$1 for large $G_2$2, via random Painter strategies and weight-function arguments (Conlon et al., 2018). Similarly, off-diagonal: $G_2$3 for fixed $G_2$4.

4. Optimal Strategies and Methodology

Builder Strategies

Builder exploits forced Painter moves by targeting edges which, if colored improperly, immediately complete the target graph. Inductive constructions, path- and cycle-extensions, and template-based case analyses dominate strategy design. For $G_2$5-vs-path, a three-stage strategy seeds "structural units," extends them into gadgets with prescribed monochromatic subpaths, and finally connects them into the desired path—all steps orchestrated to restrict Painter's freedom and extract forced blue extensions (Zhang et al., 2023, Bednarska-Bzdȩga, 2023).

For cycles, path-building methods shift to "cycle-shortening" lemmas, and for more complex pairs such as $G_2$6, Builder repeatedly forces small configurations (roots, gadgets, wavy paths) until the closure lemma applies (Zhi et al., 9 Jan 2026).

Painter Strategies

Optimal Painter play consists of deferred creation of forbidden structures, coloring red unless a red copy of $G_2$7 or closed red cycle would emerge, and blue otherwise. Defensive coloring exploits local degrees, forced star or matching avoidance, and for ordered games, left/right degree balance (Heath et al., 2024, Adamski et al., 2022).

Potential-function methods are used to quantify the growth of blue and red subgraphs, yielding tight lower bounds on total rounds by tracking parameters such as degree-1 vertices or component structure (Adamska et al., 21 Apr 2025, Mond et al., 2023).

5. Restricted, Ordered, Induced, and Turnaround Online Ramsey Numbers

Restricted Online Ramsey Numbers

For a finite host of $G_2$8 vertices,

$G_2$9

with matching upper/lower bounds achieved through refined edge-typing and matching-packing arguments (Dvořák, 2020).

Ordered Online Ramsey Numbers

In ordered settings, the online ordered Ramsey number is controlled by the interval-chromatic number and left/right degrees. Tight bounds have been established: $\tilde{r}(G_1,G_2)$0 for arbitrary ordered $\tilde{r}(G_1,G_2)$1 (Clemen et al., 2022); $\tilde{r}(G_1,G_2)$2 for $\tilde{r}(G_1,G_2)$3-ichromatic $\tilde{r}(G_1,G_2)$4. Lower bounds are governed by maximal degrees: $\tilde{r}(G_1,G_2)$5 (Heath et al., 2024).

Induced Online Ramsey Numbers

The induced variant $\tilde{r}(G_1,G_2)$6 requires Builder to force an induced monochromatic $\tilde{r}(G_1,G_2)$7. For paths and cycles, linear-in-$\tilde{r}(G_1,G_2)$8 bounds are proven: $\tilde{r}(G_1,G_2)$9 (Blažej et al., 2019). Notably, for thorn-regular caterpillars $$\tilde{r}(G_1, G_2) = \min \left\{ N : \text{Builder has a strategy to force a red } G_1 \text{ or a blue } G_2 \text{ within } N \text{ moves} \right\}$$0, one has

$$\tilde{r}(G_1, G_2) = \min \left\{ N : \text{Builder has a strategy to force a red } G_1 \text{ or a blue } G_2 \text{ within } N \text{ moves} \right\}$$1

independent of $$\tilde{r}(G_1, G_2) = \min \left\{ N : \text{Builder has a strategy to force a red } G_1 \text{ or a blue } G_2 \text{ within } N \text{ moves} \right\}$$2, establishing a gap phenomenon relative to induced size-Ramsey numbers.

Turnaround Numbers

In the 3-color turnaround game, Painter forces a monochromatic $$\tilde{r}(G_1, G_2) = \min \left\{ N : \text{Builder has a strategy to force a red } G_1 \text{ or a blue } G_2 \text{ within } N \text{ moves} \right\}$$3 while Builder restricts one color per edge. General bounds are: $$\tilde{r}(G_1, G_2) = \min \left\{ N : \text{Builder has a strategy to force a red } G_1 \text{ or a blue } G_2 \text{ within } N \text{ moves} \right\}$$4 with $$\tilde{r}(G_1, G_2) = \min \left\{ N : \text{Builder has a strategy to force a red } G_1 \text{ or a blue } G_2 \text{ within } N \text{ moves} \right\}$$5 the polychromatic extremal function (Almási et al., 8 Dec 2025). Asymptotics reveal a density threshold controlled by the chromatic number and polychromatic colorings.

6. Connections to Other Invariants and Open Problems

Online Ramsey numbers interlink with Turán numbers, polychromatic Ramsey theory, and set-coloring Ramsey numbers. Set-coloring thresholds, chromatic-Ramsey numbers, and generalized polychromatic functions appear throughout as bounding tools (Almási et al., 8 Dec 2025). For paths, the exact linear constant for $$\tilde{r}(G_1, G_2) = \min \left\{ N : \text{Builder has a strategy to force a red } G_1 \text{ or a blue } G_2 \text{ within } N \text{ moves} \right\}$$6 is resolved for $$\tilde{r}(G_1, G_2) = \min \left\{ N : \text{Builder has a strategy to force a red } G_1 \text{ or a blue } G_2 \text{ within } N \text{ moves} \right\}$$7, with $$\tilde{r}(G_1, G_2) = \min \left\{ N : \text{Builder has a strategy to force a red } G_1 \text{ or a blue } G_2 \text{ within } N \text{ moves} \right\}$$8 remaining open (Adamska et al., 21 Apr 2025, Mond et al., 2023).

For ordered online Ramsey, the gap between $$\tilde{r}(G_1, G_2) = \min \left\{ N : \text{Builder has a strategy to force a red } G_1 \text{ or a blue } G_2 \text{ within } N \text{ moves} \right\}$$9 and $G_1,G_2$0 in path-path games is a major open question; for cycles, constant-factor improvements in the odd case remain conjectural (Heath et al., 2024, Clemen et al., 2022).

In the restricted game, saving $G_1,G_2$1 moves over the naive bound $G_1,G_2$2 is now possible for the diagonal case; the off-diagonal threshold is conjectured to admit constant fraction savings (Gonzalez et al., 2018).

Gap phenomena between online, induced online, and size-Ramsey numbers continue to guide research into the efficiency of sequential graph-building under adversarial coloring (Blažej et al., 2019).