Arguably the most fundamental question in sieve theory is the following: For a set of primes P, what is the number of positive integers n < x all of whose prime factors lie in the set P. Classical sieve methods can answer this question in case the set P contains all the primes larger than x1/2. A while ago, Andrew Granville, Dimitris Koukoulopoulos and I initiated the study of what happens when one sieves out also some large primes --- we made a conjecture, related it to an additive combinatorial conjecture, and proved a weak version of it. In a very recent work Xuancheng Shao and I have managed to prove the whole conjecture. In the talk I will discuss the conjecture, how it is related to additive combinatorics and what additive combinatorial tools we use.