Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert.
I guess I could try to find something implying it in that 400-page book (called "Sieve Methods", available online in unsearchable format) but I thought I'd ask if maybe somebody knows a precise reference [or a proof!] for this result?



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> For every $\epsilon>0$, there is a constant $c=c(\epsilon)$ such that at least $c\frac{k}{\log k}$ integers between $n$ and $n+k$ have all their prime factors greater than $k^{1/2-\epsilon}$.
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Unpublished result of Rosser in Sieve Methods book
This is part of the theory of the "linear sieve." Chapter 8 of the book of Halberstam and Richert deals with this topic. Alternatively you could look at Iwaniec's paper [On the error term in the linear sieve](http://matwbn.icm.edu.pl/ksiazki/aa/aa19/aa1911.pdf). The result you want can be extracted from the lower bound (1.4) on page 2 of Iwaniec's paper, together with the formula for $f(s)$ at the top of that page.


