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Claim. For $f=\tau$, the divisor function, and for $f=\omega$, the number of distinct prime factors, the property asked in Problem 122 holds: for every $F$ with $F(n)/f(n)\to0$ for almost all $n$ and, as [Er97] requires, $F(x)\to\infty$, there are infinitely many $x$ along which the number of $n$ with $n+f(n)\in(x,x+F(x))$, divided by $F(x)$, tends to infinity. Erdős reports the result in Problems in number theory, New Zealand J. Math. 26 (1997), 155--160, p. 155, and in Some of my favourite unsolved problems, Math. Japon. 46 (1997), 527--537, p. 533: he writes that he, Pomerance and Sárközy can prove it for $\tau$ and $\omega$, and that it probably fails for $\phi$ and $\sigma$. Neither paper gives an argument and no publication of the proof is recorded; the site's commentary (page last edited 2026-04-01) repeats the report.

Covers. $f=\tau$ and $f=\omega$, in the corrected reading of the question: $F(n)/f(n)\to0$ for almost all $n$, with the width $F(x)\to\infty$ as [Er97] requires. Every other $f$, including $\phi$ and $\sigma$, for which Erdős expected the property to fail, is outside the claim.

Published results for $\omega$. The paper of Erdős, Pomerance and Sárközy on locally repeated values, part IV (card; Ramanujan J. (1997), 227--241), proves results for $f=\omega$ at single widths only. Its Theorem 1 gives, for every large $x$, some $n\le x$ with more than $c(\log x)^{1/2}(\log\log x)^{-1}$ values $m$ satisfying $m+\omega(m)=n$: clustering at a bounded width. Its method gives, as the site's commentary and the curator's thread comment of 2026-03-27 record, an interval $I\subseteq[1,O(x)]$ of width about $((\log x)/\log\log x)^{1/2}$ whose points $n$ all have $n+\omega(n)$ in one interval $J$ of width about $(\log\log x)^{1/2}$. Each result fixes one width $F$ and settles no instance of the property, which quantifies over every $F$, so neither is a claim on this problem. In the same comment the curator, Thomas Bloom, wrote that the results Erdős describes do not really appear in that paper.

Depends on. No page of this wiki.

Acceptance. None recorded. Erdős's report is the poser's word that a proof exists, with no argument to examine, and this page does not count it as acceptance evidence. The site labels the problem OPEN (page last edited 2026-04-01) and its proof-claims tab carries no entry, so the commentary repeating the report credits nothing. The curator's doubt of 2026-03-27 is not a refutation. There is no refereed proof and no formalization, so the claim stays claimed.