Erdős Problem #1041
FALSIFIABLEdatabase updated 2025-09-15
Still open — but if it is false, a finite counterexample exists, so a computer search could in principle disprove it. A search can't prove it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The length of a subset \(s\) of \(\mathbb{C}\) is defined to be its 1-dimensional Hausdorff measure \(\mathcal{H}^1(s)\).
**Erdős–Herzog–Piranian Component Lemma** (Metric Properties of Polynomials, 1958): If \(f\) is a monic degree \(n\) polynomial with all roots in the unit disk, then some connected component of \(\{z \mid |f(z)| < 1\}\) contains at least two roots with multiplicity. See p. 139, above Problem 5: [EHP58] Erdős, P. and Herzog, F. and Piranian, G., _Metric properties of polynomials_. J. Analyse Math. (1958), 125-148.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/1041.Prizenone
Topicsanalysis, polynomials
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1041
Attempts
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The live page is FALSIFIABLE/open with 0 claimed proofs, but lists “Currently working on this problem: ogroth” (accessed 2026-07-31), so the mandatory no-collision rule applies.
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