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Claim. For every monic polynomial $f\in\mathbb{C}[z]$ of degree $2$, $3$ or $4$, the set $\{z\in\mathbb{C}:\lvert f(z)\rvert\le1\}$ can be covered by disks the sum of whose radii is at most $2$. This is the question of Problem 509 restricted to degree at most four. The claim was registered on the site's proof-claims page on 2026-07-19 by Boon Qing Hong as a partial claim, with a write-up in a shared PDF, linked above, that the claimant says was mainly generated by ChatGPT (the claim's tools line names Chat GPT-5.5 Pro and 5.6 Sol). That PDF is a different document from the claimant's strategy note on the general question, carded as Hong 2026, which proposes a route and proves no case; this claim rests on the PDF linked above.

Submission note. Posted to erdosproblems.com as a proof claim by Boon Qing Hong (account Qing_Hong) on 19 July 2026, giving "Chat GPT-5.5 Pro, 5.6 Sol" as the AI used:

Here is a result on the problem up to quartic polynomials, mainly generated by ChatGPT. - The quadratic case is trivial. - The cubic case can be solved by bounding the variables. - The quartic case can be solved by bounding as well, followed by a computer-assisted proof to bound $\mathcal{F}(X, Y, Z).$ (Code is included in PDF) GPT also suggested an MST approach as a promising approach among several approaches.

Covers. Degrees $2$, $3$ and $4$ only; the claimant calls the quadratic case trivial. (Degree $1$ is not part of the claim; this page notes only that it is immediate, the set being a disk of radius $1$.) The general question, every degree, is untouched; the known general bounds are Cartan's $2e$ and Pommerenke's $2.59$ in place of $2$, and Pommerenke proved that $2$ suffices when the open set $\{|f|<1\}$ is connected, the accepted partial claim on Pommerenke 1959 (library card, cited on the problem page as [Po59]), and more generally when the closed set $\{|f|\le1\}$ is connected, the accepted partial claim on Pommerenke 1961.

Argument, as the claim summarizes it. The cubic case is settled by bounding the variables, and the quartic case by bounding as well, followed by a computer-assisted bound on a function the write-up calls $\mathcal{F}(X,Y,Z)$, with the code included in the PDF. The summary adds that the system also suggested an approach through minimum spanning trees as a promising route to the general question; that suggestion is not a result and is not covered by this claim. The write-up is behind a sign-in; this account follows the site's claim entry (2026-10-07).

Depends on. No page of this wiki.

Acceptance. None recorded. The site's label is OPEN (problem page last edited 2025-12-29), the proof-claims entry carries no comments and no acceptance mark, and no review, referee report or acknowledgment by anyone outside the claimant was found (2026-10-07).