Erdős Problem #1186

Erdős problem #1186 · erdosproblems.com/1186

What is the minimum asymptotic density $\delta_k$ of monochromatic $k$ -term arithmetic progressions in every two-colouring of $\{1, \dots, n\}$ ? The exact certificate gives $\delta_3 = 117/2192$ , matching the known 548-bead colouring.

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Argument
Field: Arithmetic Ramsey Theory
Posed by: Paul Erdos (general case); Pablo Parrilo, Aaron Robertson and Dan Saracino (specific case)
Year posed: 2008
Years open: 18y
Solved: 2026-07-13
Model: GPT-5.6 Star Fleet (Claude Fable 5 referee)
Vendor: OpenAI / Anthropic
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Significance: 10 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

exact k = 3 constant established, settling the Parrilo-Robertson-Saracino conjecture for 3-APs; general k remains open.

Verification

Dual exact checkers plus a partial Lean formalization; community status pending. Not yet mentioned on Thomas Bloom's database.

Sources

Lean proofStar Fleet's repo of the proof
Problem recorderdosproblems.com/1186Parrilo et al 2008 - Problem statement

Erdős Problem #1186

What was actually shown

Verification

Sources

Discussion