2026
PROVEN
Limits RegistryPUBLIC RECORDLimits RegistryPUBLIC RECORDLimits RegistryPUBLIC RECORDLimits RegistryResearch Console →Ramsey's theorem guarantees that large enough structures always contain order: however you two-color the edges of a large enough complete graph, a monochromatic triangle becomes unavoidable. R(3,3) is the smallest such threshold.
The question, scope, and sources behind this Registry record.
What is the least n such that every two-coloring of the edges of the complete graph K_n contains a monochromatic triangle?
Current frontiers derived from accepted Claims.
The accepted equality closes this optimization result.
Assertions tied to evidence, attribution, and review.
The frontier as it changed over time.
Only accepted Claims matching the current specification contribute to the displayed bounds. Strict inequalities remain open; contradictory Claims require editorial review.