Four Random Permutations Conjugated by an Adversary Generate $S_n$ with High Probability

We prove a conjecture dating back to a 1978 paper of D.R. Musser [Mus78], namely that four random permutations in the symmetric group Sn generate a transitive subgroup with probability pn > ε for some ε > 0 independent of n, even when an adversary is allowed to conjugate each of the four by a possibly different element of Sn (in other words, the cycle types already guarantee generation of Sn). This is closely related to the following random set model. A random set M ⊆ Z is generated by including each n ≥ 1 independently with probability 1/n. The sumset sumset(M) is formed. Then at most four independent copies of sumset(M) are needed before their mutual intersection is no longer infinite.