---

Claim. Consider the node systems in $[-1,1]$ that contain both endpoints, the paper's convention, and for such a system let $\lambda_i$ be the maximum of the Lebesgue function $\sum_k|l_k(x)|$ on the $i$-th gap between consecutive nodes, so that $\min_i\lambda_i$ is the quantity $\Upsilon$ of Problem 1130 in that convention. Theorem 2 of de Boor and Pinkus (p. 298) states that two systems $s,t$ with $\lambda_i(s)\le\lambda_i(t)$ for every $i$ are equal; hence every system satisfies

$$\min_i\lambda_i\le\lambda^*\le\max_i\lambda_i,$$

where $\lambda^*$ is the minimal Lebesgue constant, which is the conjecture Erdős stated in [Er47] and repeated in 1958 (the paper's (1)). Together with Theorem 1 (p. 295), that exactly one system $t^*$ equioscillates ($\lambda_1=\cdots=\lambda_n$), and the displayed inequality applied to $t^*$, which forces its common value to be $\lambda^*$ (the paper's Corollary on p. 295 reaches the same value through Kilgore's theorem [Ki77], that a minimizer equioscillates), this answers the second question in the paper's convention: $\Upsilon$ is maximized by $t^*$ alone, with maximum $\lambda^*$, since a system $t$ with $\lambda_i(t)\ge\lambda^*=\lambda_i(t^*)$ for every $i$ equals $t^*$ by Theorem 2. The first question follows: $\lambda^*$ is at most the Lebesgue constant of the Chebyshev nodes rescaled to span $[-1,1]$, which is at most that of the Chebyshev nodes themselves, $(2/\pi)\log n+O(1)$ by the classical computation that the site states on Problem 1129 and that Erdős [Er47] calls simple; so $\Upsilon\le\lambda^*\le(2/\pi)\log n+O(1)$, and the answer is yes. The paper itself does not estimate $\lambda^*$ in terms of $n$. The source card is de Boor and Pinkus 1978. The problem pairs a yes-or-no question, answered yes, with a request to describe the maximizing choice, which has neither a proved nor a disproved shape, so the claim value is answered; the site's label PROVED is recorded on the problem page.

Convention. The site's statement lets the nodes range over $[-1,1]$ and counts $n+1$ pieces, the gaps between consecutive nodes and the two end pieces between $x_0=-1$, $x_{n+1}=1$ and the extreme nodes, which is how Erdős [Er47] posed the question with his bound $\Upsilon<\sqrt n$ and his remark that the maximum is probably attained when all $n+1$ piece maxima are equal. The bound $\Upsilon\ll\log n$ holds in that convention as well: the fundamental polynomials are invariant under an affine change of variable, so the interior gaps of any system are the gaps of its rescaling onto $[-1,1]$, to which Theorem 2 applies, and $\Upsilon$ is at most the least of those interior gap maxima (an elementary observation recorded on this page, not taken from the paper and not independently reviewed). The characterization of the maximizers is proved in the paper's convention; for a system containing both endpoints the two end pieces of the site's statement are points, so such a system has $\Upsilon=1$ there, while an affine shrink of $t^*$ whose two end-piece maxima exceed $\lambda^*$ keeps $\Upsilon=\lambda^*$, so the equal-maxima characterization fails for free nodes. A third-party Lean 4 file in the lean-proofs repository, linked above at its pinned commit, with de Boor and Pinkus as informal authors and Codex and GPT-5.6 Sol as formal authors, declares itself a formalization of the literal free-node formulation and proves this for three nodes: its theorem upsilon_three_le gives $\Upsilon\le5/4$ for every three free nodes, and not_erdos_1130 exhibits the maximizer $(-1/2,0,1/2)$, with $\Upsilon=5/4$ and piece maxima $7,5/4,5/4,7$, which are not all equal. In the site's convention the maximizers are exactly the affine images of $t^*$ inside $[-1,1]$ whose two end-piece maxima are at least $\lambda^*$. If $\Upsilon(s)=\lambda^*$, every gap maximum of the rescaled system is at least $\lambda^*=\lambda_i(t^*)$. Theorem 2 then makes the rescaled system $t^*$, and both end pieces reach $\lambda^*$; conversely, such images have $\Upsilon=\lambda^*$. The symmetric image whose end-piece maxima equal $\lambda^*$ has all $n+1$ piece maxima equal, so the maximum is attained where Erdős [Er47] expected, but equal maxima do not characterize the maximizers. This step, like the bound above, is recorded on this page, not taken from the paper, and not independently reviewed.

Acceptance. Refereed: Journal of Approximation Theory 24 (1978), no. 4, 289–303, received 1977-04-01, in the issue dated December 1978 in the publisher's record, which dates this page. Reviewed: the site's curator, Thomas F. Bloom, labels the problem proved and records in its commentary that de Boor and Pinkus proved the characterization Erdős conjectured, and that the bound $(2/\pi)\log n+O(1)$ for $\Upsilon$ then follows from the estimates recorded on Problem 1129 (erdosproblems.com/1130, last edited 2026-01-17, accessed 2026-09-04). The theorem statements are taken from the paper itself, whose proofs are known here in outline only; the proofs are not compiled in this wiki. Formalization: the Lean 4 file linked above is neither built nor audited in this repository, so no formalized evidence is listed.

Depends on. Nothing in this wiki; the result rests on the refereed paper linked above, and the comparison with Problem 1129 is context.