  We show that for all large enough $x$ the interval
$[x,x+x^{1/2}\log^{1.39}x]$ contains numbers with a prime factor $p >
x^{18/19}.$ Our work builds on the previous works of Heath-Brown and Jia (1998)
and Jia and Liu (2000) concerning the same problem for the longer intervals
$[x,x+x^{1/2+\epsilon}].$ We also incorporate some ideas from Harman's book
`Prime-detecting sieves' (2007). The main new ingredient that we use is the
iterative argument of Matom\"aki and Radziwi{\l}{\l}(2016) for bounding
Dirichlet polynomial mean values, which is applied to obtain Type II
information. This allows us to take shorter intervals than in the
above-mentioned previous works. We have also had to develop ideas to avoid
losing any powers of $\log x$ when applying Harman's sieve method.

Number Theory