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~\cite{musserirred}, namely that four random permutations in the symmetric group ${\mathcal{S}}_n$ generate a transitive subgroup with probability $p_n>ϵ$ 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 ${\mathrm{\S }}_n$ (in other words, the cycle types already guarantee generation of ${\mathcal{S}}_n$). This is closely related to the following random set model. A random set $M\subseteq {\mathbb{Z}}^{+}$ is generated by including each $n \geq 1$independentlywithprobability$1/n$\mathrm{.}Thesumset$$\text{sumset}$(M)$ is formed. Then at most four independent copies of $\text{sumset}(M)$ are needed before their mutual intersection is no longer infinite.