We consider the problem of estimating the number Ψ( x, x x ) − Ψ( x − x β , x x ) of integers in the interval between x − x β and x having no prime factor greater than x x . We study when one can guarantee Ψ( x, x x ) − Ψ( x − x β , x x ) > 0 for large x and when one can guarantee Ψ( x, x x ) − Ψ( x − x β , x x ) ≥ c( α, β) x β for large x, for some positive constant c( α, β). In particular let f( α) be the infimum of the values of β for which for all α 1 > α we have Ψ( x, x x 1 ) − Ψ( x − x β , x x 1 ) > 0 for sufficiently large x, and let f ∗(α) be the infimum of values of β for which for all α 1 > α we have Ψ( x, x x 1 ) − Ψ( x − x β , x x 1 ) ≥ c( α 1, β) x β for some c( α 1, β) > 0 for sufficiently large x. We prove using an idea of Chebyshev that there exists a positive constant c such that, for 0 ≤ α ≤ 1, f ★(α)⩽1−α−cα(1−α) 3 By combining an elementary extrapolation technique with an explicit construction valid for α −1 = 2, 3, 4, …, we show that for 0 < α ≤ 1 2 , f(α)⩽1−2α(1−2 −[x −1] ) We also prove results implying that for fixed α, β and almost all x one has Ψ(x, x x) − Ψ(x − x β, x x) ≥ ( 1 64 ) βϱ( 1 α ) x β , where ϱ( t) is Dickman's function.