  We prove the infinitude of shifted primes $p-1$ without prime factors above
$p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998.
Consequently, we obtain an improved lower bound on the the distribution of
Carmichael numbers. Our main technical result is a new mean value theorem for
primes in arithmetic progressions to large moduli. Namely, we estimate primes
of size $x$ with quadrilinear forms of moduli up to $x^{17/32}$. This extends
moduli beyond $x^{11/21}$, recently obtained by Maynard, improving $x^{29/56}$
from well-known 1986 work of Bombieri, Friedlander, and Iwaniec.
