Erdős Problem 625 — A Proposed Proof of the Divergence of the Chromatic–Cochromatic Gap

This preprint presents a proposed proof of Erdős Problem 625: for the random graph G(n, 1/2), the difference between its chromatic and cochromatic numbers tends to infinity with high probability along the full sequence of positive integers.

The argument combines four parameter bands. Heckel’s published theorem is used for the central range, while the manuscript develops arguments for the remaining boundary ranges. The key proposed step is Proposition 6.1 (SM-CO-TAIL), a cochromatic second-moment estimate supported by the similar-regime adaptation in Section 6.2 and the middle-3 argument in Section 6.3. Sections 7, 8 and 10 apply this estimate, and Section 11 assembles the ranges.

The manuscript was developed through a multi-cycle agentic AI research workflow involving iterative derivation, source checking and adversarial model audits in fresh contexts, rather than a single-prompt generation. These checks remain model-based: the proposed proof has not yet received an independent human expert review and has not been formally verified in a proof assistant.

The manuscript is made available to invite independent mathematical scrutiny, particularly of Proposition 6.1 and Sections 6.2–6.3. The date stated on the manuscript is 8 August 2026.