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Claim. With $D(P)$ the largest deviation, over spherical caps $C$, of the count $\lvert C\cap P\rvert$ from its expectation $\alpha_C\lvert P\rvert$,

$$\min_{\lvert P\rvert=n}D(P)\to\infty\qquad(n\to\infty)$$

for finite sets $P$ on the unit sphere $S^2$, so the answer to Problem 988 is yes. Schmidt proves it for the sphere of every dimension, as the fourth part of his series on irregularities of distribution. The analogous statement for the unit square, with axis-parallel boxes in place of caps, is Roth's theorem of 1954, which the problem's sources name as the model for the question.

Depends on. Nothing in this wiki; the result rests on the cited paper alone.

Dating. The page is dated by the issue month the journal record gives (Invent. Math. 7 (1969), 55–82, March 1969); the day in the page name is a placeholder.

Acceptance. Refereed: Invent. Math. 7 (1969), 55–82. Reviewed: the site's curator, Thomas Bloom, credits Schmidt's paper with the affirmative answer in the problem's commentary (read 2026-10-07; the site's thread holds no comment and no proof claim), and the community database lists the problem as solved. The claim value is proved: the problem asks a yes-or-no question and the theorem answers it yes. The paper is not held here, so the quantitative lower bound it gives is not transcribed on this page.

Formalization. The file src/latest/ErdosProblems/Erdos988.lean of Boris Alexeev's lean-proofs repository (GitHub plby/lean-proofs), linked above at the commit read and first added, declares itself a Lean formalization of a solution to the problem, with Schmidt as its informal author and Codex and GPT-5.6 Sol as its formal authors. For Lean and Mathlib v4.33.0 it defines the spherical cap discrepancy of a finite $P\subseteq S^2$ as the supremum, over the closed caps $\{x:\langle x,u\rangle\ge t\}$ with $t\in[-1,1]$, of $\lvert\,\lvert C\cap P\rvert-\alpha_C\lvert P\rvert\,\rvert$ with $\alpha_C=(1-t)/2$, and the minimum discrepancy of $n$-point sets as an infimum; it proves card_le_512_mul_discrepancy_pow_four, that $\lvert P\rvert\le512\,D(P)^4$ for every finite $P$, and from it erdos_988, that the minimum discrepancy tends to infinity with $n$. Its route is not Schmidt's: by its header it is the elementary Stolarsky positive-kernel argument, and it cites Bilyk and Brauchart's 2025 preprint on lower bounds for the spherical cap discrepancy beside Schmidt's paper. Neither the site's indicator nor the community database points to the file. It is not built or audited here and has no fidelity review, so formalized is not listed; the evidence of this claim is the refereed paper and the curator's acceptance.