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 n |G| < d(G) ≤ 48 n |G|.

We show that the minimal base size b(G) of a finite primitive permutation group G of degree n is at most 2(log |G|/ log n) + 24. This bound is asymptotically best possible since there exists a sequence of primitive permutation groups G of degrees n such that b(G) = ⌊2(log |G|/ log n)⌉ − 2 and b(G) is unbounded. As a corollary we show that a primitive permutation group of degree n that does not contain the alternating group Alt(n) has a base of size at most max{ √ n, 25}.