Weighted dilation graphs, smooth shifted primes and totient fibers

We prove Erdős's conjecture on the largest fibers of Euler's totient function: for every ε > 0, infinitely many positive integers n have more than $n^{1-\epsilon }$ preimages. We also show that, for every fixed δ > 0, there are at least $x^{1-o(1)}$ primes p in $2x<p\le 5x$ whose predecessors have no prime factor exceeding x${}^{\delta }$.