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Claim. For every constant $c_2>0$ there is a constant $c_1=c_1(c_2)>0$ such that, for all sufficiently large $x$, there are more than $c_1\log x$ consecutive primes $p_k<p_{k+1}<\dots<p_{k+r}<x$ whose successive gaps $p_{k+i+1}-p_{k+i}$ all exceed $c_2$; since the primes increase, every two of them then differ by more than $c_2$. This is Theorem 3 of P. Erdős, On some applications of Brun's method, Acta Univ. Szeged. Sect. Sci. Math. 13 (1949), 57–63, stated there with constants $c_5$ (the gap) and $c_6=c_6(c_5)$ (the run length $[c_6\log n]$). The proof counts the pairs of consecutive primes below $n$ at distance at most $c_5$ by Schnirelmann's sieve bound, $O(n/(\log n)^2)$, and compares that count with $\pi(n)$: the small gaps are too few to break every run of $[c_6\log n]$ primes when $c_6$ is small in terms of $c_5$. The library's [[../library/primes/erdos_1949_applications_brun_s_method/_index|card for the paper]] records the theorem beside the paper's two results on the least prime in a progression.

Covers. The instances of Problem 238 with $c_1$ below a threshold that depends on $c_2$: for every $c_2>0$ the answer is yes whenever $0<c_1<c_6(c_2)$. The problem asks whether the answer is yes for every pair $c_1,c_2>0$, that is, whether the threshold can be removed; Theorem 3 does not reach that, and the site records the problem as open.

Acceptance. The paper is refereed: it appeared in the journal Acta Universitatis Szegediensis, Sectio Scientiarum Mathematicarum, volume 13 (1949), pages 57–63. The curator of erdosproblems.com, Thomas Bloom, records in the site's commentary that Erdős proved the statement for every $c_2>0$ when $c_1>0$ is small enough in terms of $c_2$; the site's label for the problem is OPEN, so that commentary credits the partial result without settling the problem, and it is not listed as acceptance evidence.

Depends on. Nothing on this wiki; the argument is the paper's own.