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Statement

Setting (pp. 1--2). The notation of Claim 3.1: blocks $S$ of a partition of the components of $\{\lvert f\rvert<1\}$ for a monic non-constant $f$, with weighted root means $c_S$ and radii $r(S)=\sup_{z\in K_S}\lvert z-c_S\rvert$.

Minimal-counterexample setup (p. 2). Let $\Pi$ be the set of admissible partitions and $R(p)=\sum_{S\in p}r(S)$. Assume, for contradiction, that $R(p)>2$ for every $p\in\Pi$. Choose $p_*\in\Pi$ minimizing $R$ and, among all minimizers, with the smallest number of blocks. Then $R(p_*)>2$.

Lemma 4.1 (Merge irreducibility, p. 2). If $A,B\in p_*$ are distinct blocks, then $r(A\cup B)>r(A)+r(B)$.

Proof pointer

P. 2. Otherwise replacing $A$ and $B$ by $A\cup B$ gives a partition with no larger radius sum and fewer blocks, against the choice of $p_*$.

Read depth

Claims checked: the setup and Lemma 4.1 were read clause by clause on the page images of the print, and the two-line proof was followed. Nothing here is independently reviewed.

Source. Boon Qing Hong, Strategy Proposal on Covering Lemniscates for Erdős Problem #509, unpublished note (2026), 11 pp.; the edition read is named on the source card.

Bears on

• Problem 509: a property of a hypothetical minimal counterexample to Claim 3.1, used for Corollary 4.3; on its own it settles no case of the problem.