# How dense can a graph be without creating a forbidden pattern?

- ID: extremal-graph-conjectures
- 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/extremal-graph-conjectures/

## Plain-language explanation

A graph is a network of points and links. Add enough links and particular patterns become unavoidable. Extremal graph theory asks where that transition occurs.

## Research contribution

The release reports results on compactness and degeneracy conjectures, associated with Erdős problems 146 and 180.

## Academic interpretation

The forbidden graph, asymptotic variable and density notion must be fixed before comparing bounds. Degeneracy constrains how sparse every subgraph can remain under successive deletion. Compactness questions concern whether restrictions from a family can be captured by finitely many members in the relevant extremal sense. These are distinct claims and should be audited separately within the released manuscript.

## Implications

Potentially improves the structural tools used to connect local constraints with global network density.

## Caveat

This dossier groups the related announcement rather than counting it as several independent verified wins. We have not reproduced the proofs.

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