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Claim. The manuscript Polynomial bounds for infinite-dimensional vector balancing by Samuel Korsky (dated 18 September 2026, ten pages) states as its Theorem 3 that there are a coloring $f:\mathbb{N}\to\{-1,1\}$ and an absolute constant $C$ with

$$\sup_{a,m\ge1}\left\lvert\sum_{j=0}^{m-1}f(a+jd)\right\rvert\le C\,d^{5/2+2\sqrt2} \qquad(d\ge1),$$

the supremum over all finite arithmetic progressions of common difference $d$. In the notation of Problem 177, which the manuscript names as the question it addresses, this is $h(d)\ll d^{5/2+2\sqrt2}$, with $5/2+2\sqrt2\approx5.33$ in place of the exponent $8+\epsilon$ of Beck's bound $h(d)\le d^{8+\epsilon}$ for every $\epsilon>0$ [Be17], which the site's commentary records. By the manuscript's introduction and Section 5.2, the proof follows Beck's enumeration of the residue classes modulo $d$ by increasing modulus, writes a progression as the difference of two prefixes of one class, and applies the manuscript's weighted balancing theorem (its Corollary 5) to the membership vectors of the classes; the weight exploits that an integer lies in exactly one class for each modulus, and the manuscript says this saves a factor $d^{1/2}$ over applying its unweighted result, which would give the exponent $3+2\sqrt2$. The manuscript's first result, the bound $\ll d^{3/2+\sqrt2}$ for the balancing question of Problem 178, has its own page, Korsky 2026; this page records only the arithmetic-progression coloring.

Submission note. Posted to erdosproblems.com as a proof claim by Samuel Korsky (account SamKorsky) on 19 September 2026, giving "GPT Astra" as the AI used:

The paper proves that every sequence $v_n\in[-1,1]^{\mathbb N}$ admits a single signing satisfying

$$\sup_m\left|\sum_{n\le m}\delta_n v_n(k)\right| > =O(k^{3/2+\sqrt2})$$

for every coordinate $k$, improving Beck’s exponent

$4+\varepsilon$. The proof retains Beck’s recursive grouping and compactness argument but strengthens the finite cancellation step. Beck uses pigeonhole cancellation on the early coordinates and bounds the remaining coordinates by group size. Here an entropy partial-coloring argument, after weighting the coordinates, merges a positive proportion of $K$ vectors of Hilbert norm at most $B$, obtaining norm $O(\sqrt K B)$ and individual later-coordinate bounds $O(B)$. Propagating these two bounds separately and optimizing the grouping scales yields the improved exponent. Notes: We also construct a coloring of the integers with discrepancy $O(d^{5/2+2\sqrt2})$ on every finite arithmetic progression of common difference $d$, improving Beck’s $8+\varepsilon$ in Problem #177.

Covers. Only the upper bound $h(d)\ll d^{5/2+2\sqrt2}$. The problem asks for the smallest $h(d)$; the manuscript claims no lower bound, and notes that Roth's argument [Ro64], which the site records as $h(d)\gg d^{1/2}$, forces the exponent of any polynomial bound to be at least $1/2$.

Depends on. Nothing in this wiki; the manuscript's own argument carries the claim.

Standing. Claimed. The result was submitted to the site's proof-claim tab of Problem 178 on 19 September 2026, as the second result of that submission, by Samuel Korsky; the manuscript's acknowledgment says that the mathematical insights behind its improvements were generated by GPT Astra, the system the site's tab also names, and that the author takes responsibility for the content. The proof-claim tab of Problem 177 itself was empty and its label OPEN on 2026-10-07. The manuscript is hosted on a file-sharing service, is not on arXiv and has no refereed publication; the tab carries no comment on it, the site says that listing a proof claim is no guarantee of correctness and implies no examination by anyone associated with it, and no review of the manuscript is known to this corpus. The statements of Theorem 3 and Corollary 5 are checked against the manuscript; the proof is not checked by this corpus, and nothing is independently reviewed by this project.