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Claim. P. Erdős, Számelméleti megjegyzések, V. Extremális problémák a számelméletben, II (Remarks on number theory, V. Extremal problems in number theory, II), Mat. Lapok 17 (1966), 135--155, cited as [Er66] on the problem page. Section I.9 (printed p. 137) reports that Cantor, Schreiber, Straus and Erdős independently constructed a function $\varphi:\mathbb N\to\{-1,1\}$ such that for every fixed $a$ and $d$ the partial sums $\sum_{k\le m}\varphi(a+kd)$ are bounded in $m$, so that $l(a,d)=\sup_m\lvert\sum_{k\le m}\varphi(a+kd)\rvert$ is finite, using the antisymmetry $\varphi(u)=-\varphi(m+u)$; that the numbers $l(a,d)$ cannot be bounded uniformly; and that for $L(d)=\max_al(a,d)$ the example, worked out, gives the upper bound $L(d)<c^dd!$, with no good lower bound known. In the notation of Problem 177 this is $h(d)<c^dd!$, so $h(d)$ is finite for every $d$. The paper prints no proof beyond the antisymmetry remark. The source is carded at erdos_1966_szamelmeleti_megjegyzesek.

Covers. The upper bound $h(d)<c^dd!$ alone, and with it the existence of the function the problem asks about. The site's commentary records the bound as $h(d)\ll d!$, which drops the factor $c^d$. The result settles nothing about the order of $h(d)$, which the problem asks for; Beck's and Korsky's polynomial bounds, on their own claim pages, supersede it.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper is a journal publication in Matematikai Lapok, volume 17 (1966), the refereed evidence; the volume carries no month or day, so this page is dated to the first day of that year. The site's curator credits the bound to [Er66] in the problem's commentary, but the site labels the problem OPEN, so that credit is not reviewed evidence. The statement is as printed on p. 137; the paper gives no proof of the bound to check, and nothing is independently reviewed by this project.