Pyber's base size conjecture

T. Burness

Let \(G\) be a permutation group on a set \(X\). A subset \(B\subseteq X\) is a base for \(G\) if the pointwise stabilizer of \(B\) in \(G\) is trivial. The base size of \(G\), denoted \(b(G)\), is the smallest size of a base for \(G\). A well known conjecture of Pyber from the early 1990s asserts that there is an absolute constant \(c\) such that \[b(G)\leqslant\, c \frac{\mathrm{log} |G|}{\mathrm{log}\, n}\] for any primitive group \(G\) of degree \(n\). Several special cases have been verified in recent years, and I will report on recent joint work with Akos Seress that establishes the conjecture for all non-affine groups.