On Pyber’s base size conjecture
Timothy C. Burness, Ákos Seress · Transactions of the American Mathematical Society · 2015
Let $G$ be a permutation group on a finite set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer in $G$ is trivial. The base size of $G$, denoted $b(G)$, is the smallest size of a base. A well-known conjecture of Pyber from the early 1990s asserts that there exists an absolute constant $c$ such that $b(G) \leqslant c\log |G|/\log n$ for any primitive permutation group $G$ of degree $n$. Several special cases have been verified in recent years, including the almost simple and diagonal cases. In this paper, we prove Pyber’s conjecture for all non-affine primitive groups.