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Claim. Theorem 1 of P. Erdős, On some applications of Brun's method, Acta Univ. Szeged. Sect. Sci. Math. 13 (1949), 57–63 (card, Theorem 1): there are $c_1>0$ and infinitely many $d$ for which $p(a,d)>(1+c_1)\phi(d)\log d$ holds for more than $c_2\phi(d)$ reduced residues $a$ modulo $d$. The proof takes $c_1=c_2=\delta^{20}$ for a small fixed $\delta>0$. It finds such a modulus in every interval $[n,2n]$ for large $n$, but it does not identify one.

Covers. The assertion of Problem 971 along an infinite sequence of moduli, with $c=C=c_1$; not the assertion for all large $d$. The formal-conjectures file states it as the solved variant erdos_971.variants.infinite_sequence. Theorem 2 of the paper is the complementary bound $p(a,d)<\varepsilon\phi(d)\log d$ for $\gg_\varepsilon\phi(d)$ residues, and it settles no part of the question.

Acceptance. A refereed journal publication (refereed). The site labels the problem OPEN, so its commentary is not acceptance.

Depends on. No page of this wiki.