A proof of Pybers 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 $log |G| / log n leq b(G) < 45 (log |G| / log n) + c$. This finishes the proof of Pybers base size conjecture. 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 $sqrt[n]{|G|} < d(G) leq 48 sqrt[n]{|G|}$.