# How long can colors hide a monochromatic triangle?

- ID: multicolor-ramsey-triangles
- Provider: OpenAI
- Published: 2026-08-01
- Editorial check: 2026-10-29
- Status: formalized-provider-reported
- Evidence: Provider announcement; formal certificates reported, not rerun here
- Source: https://openai.com/index/ten-advances-in-mathematics/
- Public article: https://appliedfrontier.nl/en/discoveries/multicolor-ramsey-triangles/

## Plain-language explanation

Connect every pair of people and give each connection a color. How large can the group become before some three people have all three connections in the same color?

## Research contribution

The announcement reports a superexponential lower bound for multicolor triangle Ramsey numbers, linked to Erdős problem 183.

## Academic interpretation

A lower bound comes from constructing or proving the existence of colorings that avoid the forbidden triangle up to a certain size. It must not be confused with an upper bound forcing a triangle. Here the growth parameter is the number of colors. The quantifiers, asymptotic regime and exact meaning of superexponential growth matter for comparison with earlier work.

## Implications

Could substantially change expectations about how slowly unavoidable patterns emerge as additional colors are allowed.

## Caveat

A stronger lower bound is not an exact formula for every Ramsey number. The proof is source-reported and has not been reproduced here.

This is an editorial summary, NOT an independent mathematical proof check.