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Claim. For a monic polynomial $f$ of degree $n$ with all zeros in the closed unit disc, let $S(n)$ be the largest, over all such $f$, of the shortest length of a path in $E=\{z:\lvert f(z)\rvert\leq 1\}$ joining $0$ to the unit circle, the worst case that Problem 1120 asks about. The preprint of V. S. Pendyala, Shortest paths in polynomial lemniscate sublevel sets and a problem of Erdős (arXiv:2606.19178), asserts that $c\sqrt{\log n}\leq S(n)\leq \pi n$ for all sufficiently large $n$, with an absolute constant $c>0$. The preprint works in $E\cap\{\lvert z\rvert\leq 1\}$, which gives the same lengths, because a shortest path stays in the closed disc until it first meets the circle. Its abstract names the methods: an explicit geometric maze for the lower bound, Green-function and Faber-polynomial estimates, a quantization of measures on the circle, and a sweeping argument for the upper bound. See the library card.

Submission note. Posted to the site's forum by Venkata Siddharth Pendyala on 18 June 2026:

I have proved Erdős’s conjecture for this problem that the extremal shortest path length tends to infinity with $n$, but not too fast. More precisely, if $S(n)$ denotes the largest possible shortest length of a path in

$$> E_f={z\in\mathbb C: |z|\le 1,\ |f(z)|\le 1} >$$

joining $0$ to $\partial\mathbb D$, over all monic degree-$n$ polynomials whose zeros lie in $\overline{\mathbb D}$, then for all sufficiently large $n$,

$$> c\sqrt{\log n}\le S(n)\le \pi n >$$

for an absolute constant $c>0$. In particular, $S(n)\to\infty$, while the sharp asymptotic order remains open. The paper for this is available as an arXiv preprint at: https://arxiv.org/abs/2606.19178

A secondary paper I have written studies the length-one extreme of Erdős’s lemniscate path problem. While the main paper asks how long the shortest escape path in $E_f$ can be, this note asks when the optimal path can be a straight radius. I prove that every admissible polynomial of degree at most $3$ has such radial access, but give a certified degree-$6$ example for which every radius is blocked. Thus the first degree where Erdős’s path problem can fail to have a length-one extremal escape lies between $4$ and $6$.

This secondary paper is available as an SSRN preprint at: https://dx.doi.org/10.2139/ssrn.6850818

Covers. $S(n)\to\infty$, the unboundedness that Erdős presumed, together with the bound $S(n)\leq \pi n$. It does not determine the order of $S(n)$, which the problem asks for, and the abstract claims only the unboundedness of Erdős's presumption.

Standing. A single-author preprint, not refereed and with no outside review. The author announced it in the problem's thread on 2026-06-18 with the radial-access note. Asked in the thread whether AI had assisted, the author replied that the results came from about four months of their own work. The site labels the problem OPEN.

Depends on. No page of this wiki.