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Claim. For every $0<l<4$ and every $k\ge1$ there is a monic polynomial $f\in\mathbb C[z]$ such that $E=\{z:\lvert f(z)\rvert\le1\}$ has at least $k$ distinct components of diameter at least $l$. This is Theorem 1 (p. 98) of Ch. Pommerenke, On metric properties of complex polynomials, Michigan Math. J. 8 (1961), no. 2, 97–115, recorded with its one-paragraph proof on the result page. The paper states it as the negative answer to Problems 8 and 9 of Erdős, Herzog and Piranian (1958), from which Problem 511 descends. The problem asks, for every $c>1$, whether $\{z:\lvert f(z)\rvert<1\}$ has $O_c(1)$ components of diameter above $c$ independently of the degree. For any $1<c<4$ choose $l$ with $c<l<4$: the proof places $k$ disjoint segments of length $l$ in the interior of $E$, which is the open set, so the open set has at least $k$ components of diameter at least $l>c$, and no bound independent of $n$ exists. The answer to the question as posed is therefore no. The range $l<4$ cannot be enlarged, since by Pólya's theorem no component has diameter above $4$.

Acceptance. The paper is a refereed journal publication, received 26 November 1960, the refereed evidence; the publisher's record gives the year 1961 and no month or day, and this page is dated to the first day of that year. The site's curator, Thomas Bloom, labels the problem disproved and credits the negative answer to this paper, the reviewed evidence; the site also records Huang's 2025 independent rediscovery, which has its own claim page. The result page notes that the proof quotes the approximation theorem at capacity one and applies it to a set of smaller capacity without the rescaling spelled out; the note is not a review verdict and awards nothing by this corpus.

Depends on. Nothing beyond the cited paper.