Induced Ramsey Number: Theory & Bounds

• Induced Ramsey numbers are defined by requiring that every edge-coloring of a host graph yields a monochromatic induced subgraph, formalized through both two-color and multicolor formulations.
• Researchers have achieved exponential and quadratic bounds using probabilistic constructions, pseudorandom frameworks, and hypergraph container methods, resolving longstanding conjectures.
• Applications across sparse graph classes, fans, stars, and hypergraphs demonstrate how induced structural constraints lead to sharp lower bounds and efficient embedding results.

The induced Ramsey number is a Ramsey-theoretic parameter in which the sought monochromatic copy must appear as an induced subgraph of the host graph. In one common two-color formulation, a graph $F$ strongly arrows a pair $(G,H)$ if every red/blue coloring of $E(F)$ contains either a red induced copy of $G$ or a blue induced copy of $H$; the induced Ramsey number $IR(G,H)$ is the minimum order of such an $F$ (Gorgol, 2017). In the one-graph formulation, $r_{\mathrm{ind}}(H)$ is the least $N$ such that there exists a graph $G$ on $(G,H)$0 vertices for which every two-coloring of $(G,H)$1 contains an induced monochromatic copy of $(G,H)$2 (Conlon et al., 2010). A multicolor version $(G,H)$3 replaces two colors by $(G,H)$4 colors (Aragão et al., 26 Sep 2025). The subject lies at the intersection of classical Ramsey theory, pseudorandomness, probabilistic constructions, regularity, and container methods, and it now includes sharp lower bounds for structured graph pairs, near-linear or linear bounds for special sparse families, and exponential general upper bounds (Gorgol, 2017, Hunter et al., 2024, Aragão et al., 26 Sep 2025).

1. Definitions, formulations, and notation

The basic two-graph definition used in several papers is the following. A graph $(G,H)$5 strongly arrows $(G,H)$6, written $(G,H)$7, if every red/blue coloring of $(G,H)$8 yields either a red induced $(G,H)$9 or a blue induced $E(F)$0. Then

$E(F)$1

This is the formulation emphasized in work on lower bounds, multiple copies, and explicit asymmetric pairs (Gorgol, 2017, Axenovich et al., 2018, 2002.01297).

A second formulation fixes a single graph $E(F)$2. The induced Ramsey number $E(F)$3 is the least $E(F)$4 such that there exists a graph $E(F)$5 on $E(F)$6 vertices with the property that every two-coloring of $E(F)$7 contains an induced monochromatic copy of $E(F)$8 (Conlon et al., 2010). The multicolor extension is

$E(F)$9

the minimum $G$0 such that there exists a graph on $G$1 vertices whose every $G$2-edge-coloring contains a monochromatic induced copy of $G$3 (Aragão et al., 26 Sep 2025).

The literature also studies edge-minimal host graphs. For a graph $G$4 and integer $G$5, the induced $G$6-color size-Ramsey number

$G$7

is the minimum number of edges of a graph $G$8 such that every $G$9-edge-coloring of $H$0 contains a monochromatic induced copy of $H$1 (Hunter et al., 2024). Online variants replace the static host graph by a Builder–Painter game, producing the induced online Ramsey number $H$2 (Blažej et al., 2019). Hypergraph analogues are defined similarly for $H$3-uniform hypergraphs (Conlon et al., 2016).

This multiplicity of notations reflects genuinely different regimes: vertex-minimal host graphs, edge-minimal host graphs, fixed asymmetric pairs, multicolor variants, online forcing, and hypergraph generalizations. A plausible implication is that the field is best understood as a family of induced monochromatic embedding problems rather than a single scalar invariant.

2. General existence and asymptotic upper bounds

The existence of induced Ramsey numbers for graphs had been proved earlier by Deuber, Erdős–Hajnal–Pósa, and Rödl, but the main quantitative problem concerned the growth rate of $H$4 as a function of $H$5 (Conlon et al., 2010). Erdős conjectured that there exists a constant $H$6 such that every $H$7-vertex graph $H$8 satisfies

$H$9

Before 2010, the best general bound cited in the data was that Kohayakawa, Prömel, and Rödl showed

$IR(G,H)$0

using a random graph construction based on finite projective planes (Conlon et al., 2010).

A major improvement was obtained by Conlon, Fox, and Sudakov, who proved that there exists a constant $IR(G,H)$1 such that every graph $IR(G,H)$2 with $IR(G,H)$3 vertices satisfies

$IR(G,H)$4

Their method used $IR(G,H)$5-pseudorandom graphs, color-symmetric embedding lemmas, bi-density arguments, and Lovász’s partitioning lemma (Conlon et al., 2010). The paper also stated a pseudorandom host criterion: for suitable constants, every $IR(G,H)$6-vertex graph $IR(G,H)$7 appears as an induced monochromatic copy in every two-coloring of a $IR(G,H)$8-pseudorandom graph $IR(G,H)$9 on $F$0 vertices with $F$1 and $F$2 (Conlon et al., 2010).

A further step was achieved in the multicolor setting. For every graph $F$3 on $F$4 vertices and every $F$5, there exists a constant $F$6 such that

$F$7

In particular, for $F$8,

$F$9

which resolves the Erdős conjecture from 1975 in the affirmative (Aragão et al., 26 Sep 2025). The proof uses random host graphs $r_{\mathrm{ind}}(H)$0, a tailored induction, and a refined container framework for global properties, including a “novel global container lemma” and the Janson property for hypergraphs encoding induced copies (Aragão et al., 26 Sep 2025).

The quantitative development is summarized below.

Regime	General upper bound	Source
Graphs, two colors	$r_{\mathrm{ind}}(H)$1	(Conlon et al., 2010)
Graphs, $r_{\mathrm{ind}}(H)$2 colors	$r_{\mathrm{ind}}(H)$3	(Aragão et al., 26 Sep 2025)
$r_{\mathrm{ind}}(H)$4-uniform hypergraphs	$r_{\mathrm{ind}}(H)$5	(Conlon et al., 2016)

This progression shows that the general graph case moved from superexponential bounds in $r_{\mathrm{ind}}(H)$6 to exponential bounds, while the hypergraph setting remains tower-type in general (Conlon et al., 2016, Aragão et al., 26 Sep 2025).

3. Lower bounds and sharpness for graph pairs

General upper bounds are complemented by structural lower bounds for asymmetric pairs $r_{\mathrm{ind}}(H)$7. If $r_{\mathrm{ind}}(H)$8 is connected with independence number $r_{\mathrm{ind}}(H)$9 and $N$0 has clique number $N$1, then

$N$2

The paper describes this as asymptotically roughly $N$3 (Gorgol, 2017). The proof is inductive and proceeds by explicit colorings that avoid both a red induced $N$4 and a blue induced $N$5, repeatedly decomposing the host into red cliques and coloring the rest blue (Gorgol, 2017).

This bound is sharp for stars versus complete graphs: $N$6 The same paper also states that the bound is nearly sharp for paths and cycles (Gorgol, 2017).

When $N$7 is isolates-free but not necessarily connected, with independence number $N$8, and $N$9 has clique number $G$0, one has the linear lower bound

$G$1

For $G$2, the paper first proves $G$3, then uses induction for higher $G$4 (Gorgol, 2017). This lower bound is sharp for matchings: $G$5 matching the $G$6 lower bound (Gorgol, 2017).

These results show that the induced setting can force distinctly different asymptotics from classical Ramsey lower bounds. The data explicitly contrasts them with the classical Chvátal–Harary lower bound $G$7, which is typically linear, whereas the induced setting can exhibit quadratic dependence through the interaction of $G$8 and $G$9 (Gorgol, 2017).

4. Exact values and explicit families

A large part of the literature concerns explicit graph pairs for which induced Ramsey numbers can be computed exactly or confined to narrow ranges.

For stars versus a fixed graph, let $(G,H)$00. Then for every $(G,H)$01 and sufficiently large $(G,H)$02,

$(G,H)$03

The lower bound is asymptotically tight for any fixed bipartite $(G,H)$04, and the upper bound is attained up to a constant factor, for example by a clique $(G,H)$05 (2002.01297). The lower bound comes from partitioning the vertex set into $(G,H)$06 parts, coloring within-part edges blue and between-part edges red, thereby forbidding both a red induced $(G,H)$07 and a blue induced $(G,H)$08 (2002.01297). For stars on both sides one has the exact formula

$(G,H)$09

[Lemma 4, (2002.01297)].

For fan graphs $(G,H)$10, defined as the union of $(G,H)$11 triangles sharing one central vertex, there is a quadratic upper bound: for every fixed graph $(G,H)$12, there exists a constant $(G,H)$13 such that

$(G,H)$14

This is proved using finite projective planes, random embeddings of a Ramsey graph for $(G,H)$15, the Caro–Wei lemma, and a conflict analysis tailored to induced copies of the fan (Zhong et al., 20 Mar 2026). When $(G,H)$16 is a star $(G,H)$17 with $(G,H)$18,

$(G,H)$19

and for $(G,H)$20,

$(G,H)$21

(Zhong et al., 20 Mar 2026).

Multiple-copy phenomena show that induced Ramsey numbers are not simply additive. If $(G,H)$22 denotes the disjoint union of $(G,H)$23 copies of $(G,H)$24, then

$(G,H)$25

This upper bound comes from taking $(G,H)$26 disjoint copies of a host graph that arrows $(G,H)$27 (Axenovich et al., 2018). For many known examples equality holds, including

$(G,H)$28

and

$(G,H)$29

(Axenovich et al., 2018). However, strict inequality can occur: for $(G,H)$30,

$(G,H)$31

and the gap can grow with $(G,H)$32 (Axenovich et al., 2018). This suggests that induced structure can create efficiencies unavailable in the naive disjoint-union argument.

5. Induced size-Ramsey numbers and sparse graph classes

The edge-minimal analogue, the induced size-Ramsey number, asks for the minimum number of edges in a host graph that forces a monochromatic induced copy under every coloring. For fixed integers $(G,H)$33, if $(G,H)$34 is an $(G,H)$35-vertex graph with maximum degree $(G,H)$36 and treewidth at most $(G,H)$37, then

$(G,H)$38

This linear bound covers all bounded-degree graphs of bounded treewidth, including trees and cycles, and is proved via a “novel reduction argument” that converts non-induced size-Ramsey constructions into induced ones through a pseudo-random blowup, cleaning, and embedding procedure (Hunter et al., 2024).

For cycles, more explicit multicolor bounds are known. For all sufficiently large $(G,H)$39,

$(G,H)$40

when $(G,H)$41 is even, and

$(G,H)$42

when $(G,H)$43 is odd (Bradač et al., 2023). The odd-cycle result is close to the lower bound $(G,H)$44, while the even-cycle result replaces the older tower-type dependence on $(G,H)$45 by a polynomial one (Bradač et al., 2023). The proof uses gadget graphs, random hypergraphs, local sparsity, expansion, and DFS-based methods to avoid the regularity lemma (Bradač et al., 2023).

Long subdivisions of bounded-degree graphs also admit explicit linear induced size-Ramsey bounds. If $(G,H)$46 has maximum degree $(G,H)$47, $(G,H)$48 is a subdivision in which every replaced edge has length greater than $(G,H)$49, and $(G,H)$50, then

$(G,H)$51

If every subdivision length is even and larger than $(G,H)$52, this improves to

$(G,H)$53

(Javadi et al., 5 Feb 2026). The methods use sparse high-girth hypergraphs, gadget graphs for short cycles, expansion, and “reserved critical vertices,” explicitly avoiding Szemerédi’s regularity lemma (Javadi et al., 5 Feb 2026).

These sparse-host results show that induced Ramsey forcing can be linear in $(G,H)$54 for broad structured classes even when the general vertex Ramsey problem is much harder.

6. Extensions: online and hypergraph settings, and structural context

The induced online Ramsey number $(G,H)$55 is defined in a Builder–Painter game on an infinite independent set of vertices: Builder presents edges one by one, Painter colors them red or blue, and Builder seeks to force an induced monochromatic copy of $(G,H)$56 after finitely many rounds (Blažej et al., 2019). For paths and cycles the bounds are linear: $(G,H)$57

$(G,H)$58

and

$(G,H)$59

(Blažej et al., 2019). For spiders $(G,H)$60,

$(G,H)$61

while for centipedes $(G,H)$62,

$(G,H)$63

and there exists an infinite family of trees $(G,H)$64 such that

$(G,H)$65

(Blažej et al., 2019). These comparisons show that online and size-based induced parameters can behave quite differently.

Hypergraph induced Ramsey numbers exhibit another direction of generalization. For a $(G,H)$66-uniform hypergraph $(G,H)$67, the induced Ramsey number $(G,H)$68 is the smallest $(G,H)$69 such that some $(G,H)$70-uniform hypergraph $(G,H)$71 on $(G,H)$72 vertices has the property that every two-coloring of $(G,H)$73 contains an induced monochromatic copy of $(G,H)$74 (Conlon et al., 2016). The main theorem bounds $(G,H)$75 by a reasonable power of the classical hypergraph Ramsey number $(G,H)$76, and in particular implies that for every $(G,H)$77-uniform hypergraph $(G,H)$78 on $(G,H)$79 vertices,

$(G,H)$80

for some constant $(G,H)$81 (Conlon et al., 2016). The proof relies on the hypergraph container method (Conlon et al., 2016).

A related structural theme appears in work on Ramsey graphs. Every $(G,H)$82-vertex $(G,H)$83-Ramsey graph contains at least $(G,H)$84 induced subgraphs with pairwise distinct $(G,H)$85 pairs, proving the Erdős–Faudree–Sós conjecture (Kwan et al., 2017). Although this is not itself an induced Ramsey-number theorem, it reinforces the recurring principle that Ramsey-type host graphs exhibit strong induced-subgraph richness (Kwan et al., 2017).

7. Methods, themes, and open directions

Several proof paradigms recur across the subject. Explicit coloring constructions yield lower bounds by forbidding red induced $(G,H)$86 through control of independence structure and blue induced $(G,H)$87 through clique obstructions or degree bounds (Gorgol, 2017, 2002.01297). Pseudorandom graphs support dense general upper bounds, as in the $(G,H)$88-pseudorandom framework used to prove $(G,H)$89 (Conlon et al., 2010). Random host graphs and hypergraph containers underpin the exponential multicolor upper bound $(G,H)$90 (Aragão et al., 26 Sep 2025). Sparse induced size-Ramsey constructions rely on blowups, gadgets, expansion, local sparsity, and cleaning procedures (Hunter et al., 2024, Bradač et al., 2023, Javadi et al., 5 Feb 2026). Hypergraph container methods also extend the theory beyond graphs (Conlon et al., 2016).

Several open problems are explicitly recorded in the data. Before the 2025 exponential upper bound, the main unresolved question for graphs was Erdős’s conjecture $(G,H)$91 (Conlon et al., 2010); the multicolor paper states that this is now resolved for $(G,H)$92, and more generally proves $(G,H)$93 (Aragão et al., 26 Sep 2025). In induced size-Ramsey theory, the “Absolute Constant Problem for Trees” asks whether $(G,H)$94 can hold for all $(G,H)$95-vertex trees with a universal constant independent of the maximum degree, though the data also notes that known lower bounds show some high-degree trees require much larger host graphs (Hunter et al., 2024). For fans, it remains open whether the quadratic upper bound $(G,H)$96 is asymptotically tight, and whether analogous methods extend to fans built from higher cycles (Zhong et al., 20 Mar 2026). For cycles and long subdivisions, improving the dependence on the number of colors $(G,H)$97 remains an explicit quantitative theme (Bradač et al., 2023, Javadi et al., 5 Feb 2026).

Taken together, these results indicate that induced Ramsey theory has two contrasting faces. On one side, the unrestricted vertex Ramsey problem for arbitrary graphs requires sophisticated probabilistic and container methods and attains exponential-scale host sizes (Aragão et al., 26 Sep 2025). On the other, substantial structured classes—trees, bounded-treewidth graphs, cycles, matchings, long subdivisions, stars against fixed targets, and fans—admit linear, quadratic, or exact bounds driven by combinatorial structure (Gorgol, 2017, 2002.01297, Hunter et al., 2024, Bradač et al., 2023, Zhong et al., 20 Mar 2026). This suggests that the central organizing principle of the subject is the tension between global universality and local induced constraints.