Most primitive groups are full automorphism groups of edge-transitive hypergraphs

TL;DR

The paper demonstrates that most primitive groups, excluding the alternating group, are the full automorphism groups of certain edge-transitive hypergraphs, with high probability as the set size grows, and provides bounds on hypergraph edge sizes.

Contribution

It proves that all primitive groups except the alternating group are likely automorphism groups of edge-transitive hypergraphs, answering a longstanding question and establishing optimal bounds on edge sizes.

Findings

Probability tends to 1 that Aut(X,Y^G) = G for random hypergraph edges.

All primitive groups except the alternating group are automorphism groups of hypergraphs.

Provides an essentially optimal upper bound on minimal edge size in such hypergraphs.

Abstract

We prove that, for a primitive permutation group G acting on a set of size n, other than the alternating group, the probability that Aut(X,Y^G) = G for a random subset Y of X, tends to 1 as n tends to infinity. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M. Klin. Moreover, we give an upper bound n^{1/2+\epsilon} for the minimum size of the edges in such a hypergraph. This is essentially best possible.