Extremal graph theory · induced subgraphs · Ramsey-type homogeneous sets

Erdős–Hajnal Conjecture

Collaboration beta

If a large graph avoids one fixed induced pattern, must it contain a clique or independent set whose size is a fixed positive power of the number of vertices? The general conjecture remains open.

hom (G) = \max {α (G), ω (G)} \geq n^{δ_{H}}

Listed inErdős Problems #61

Known results and sources

The conjecture asks whether forbidding any one fixed induced graph forces a polynomial-size clique or stable set; the general case is open.

Research problem

Exact mathematical statement

For a finite graph G, write $\operatorname{hom}(G)=\max\{\alpha(G),\omega(G)\}$ . For every finite graph $H$ , the Erdős–Hajnal conjecture asks whether there exists $\delta_H>0$ such that every $n$ -vertex induced- $H$ -free graph $G$ satisfies

\operatorname{hom}(G)\ge n^{\delta_H}.

Problem infographic

Problem at a glance

A four-part landscape fixes a graph H, forbids it as an induced subgraph of G, compares a clique with a stable set, and marks the every-H frontier open. — The explainer separates the exact graph objects and homogeneous-set bound from the special-case frontier: all five-vertex H are known, but arbitrary H remains open.

Current mathematical picture

Where work on Erdős–Hajnal Conjecture stands

Open conjecture

The source reports selector-safe conditional routes into the retained recurrence and a refined seven-part frontier. The full conjecture and exact-threshold foundation remain unproved and independently uncertified.

Leading routeSelector-safe recurrence routes with unresolved ambient termination

The source retains rooted-deck and broad polar routes into the weighted recurrence, while keeping decorated-spine termination, owner synchronization, near-boundary polar states, legacy depth, and exact-threshold certification open.

Route status · Active route

Useful failureReusing previously superseded diffuse heuristics or treating local structure as a global module

The source's ten-item ledger records the controlling failures and warns that the unarchived nine-vertex computation is not exact evidence. The source keeps a conditional weighted-conflict and frozen-block architecture viable only if the m-way collar, Bregman comparison, critical-floor selection, and later globalization all survive audit.

Route status · Eliminated route

Main reductionExact polynomial homogeneous-set target

The governing source states the standard polynomial lower bound for the maximum of clique number and independence number in every induced-H-free graph.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeWO57-F — exact-threshold certification

Independently rebuild: - the critical obstruction and quantifier order; - full support, atom cap, and primeness; - exact substitution and modular maximum; - cleanup with its correct additive defect; - pair and block switch-response identities; - the threshold entropy/Gibbs collar; - and the final conversion to an ordinary Erdős–Hajnal exponent.

Task status · Ready to work on

Later mathematical updateSource reports a repaired conditional route into the retained recurrence

The v57 source reports selector-safe rooted-deck and polar routes into the retained recurrence while keeping every endpoint conditional.

v57 source revision order; not a claim of occurrence time

Work mapped so far

Erdős–Hajnal Conjecture in numbers

24.1kretained lines of mathematical investigation24,070 in the current working snapshot

Argument development: 20,725 · 86%
Explored or eliminated routes: 687 · 3%
Computational analysis: 607 · 3%
Open obligations: 851 · 4%
Definitions and setup: 1,237 · 5%

12selected mapped statements2routes investigated7open questions7contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byPaul Erdős and András HajnalProposed1989Status checkedAug 14, 2026

Source recordedAug 21, 2026

Recommended next task

WO57-F — exact-threshold certification

Suggested move: Independently rebuild: - the critical obstruction and quantifier order; - full support, atom cap, and primeness; - exact substitution and modular maximum; - cleanup with its correct additive defect; - pair and block switch-response identities; - the threshold entropy/Gibbs collar; - and the final conversion to an ordinary Erdős–Hajnal exponent.

What would count as progress

Supply an exact source-independent argument or scoped counterexample with every imported premise identified.
Survive a separate mathematical review of statement scope and dependency closure.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

22 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeSelector-safe recurrence routes with unresolved ambient termination

Route status · Active route

Explored alternatives

Other routes

1 recorded

Eliminated routeReusing previously superseded diffuse heuristics or treating local structure as a global module

Route status · Eliminated route

Route statements and reductions

Statements the next route can inspect and build on

Route statementv57 conditional recurrence reduction

The source reports that the rooted physical-deck non-depth route and a broad dual-core polar range feed actual selector-pair families into the retained weighted recurrence; endpoints remain a global module or one of two unresolved spines.