A proof of Pyber's base size conjecture

Building on earlier papers of several authors, we establish that there exists a universal constant c>0 such that the minimal base size b(G) of a primitive permutation group G of degree n satisfies log⁡|G|/log⁡n≤b(G)<45(log⁡|G|/log⁡n)+c. This finishes the proof of Pyber's base size conjecture. The main part of our paper is to prove this statement for affine permutation groups G=V⋊H where H≤GL(V) is an imprimitive linear group. An ingredient of the proof is that for the distinguishing number d(G) (in the sense of Albertson and Collins) of a transitive permutation group G of degree n>1 we have the estimates |G|n