  Using data samples collected with the BESIII detector at the BEPCII collider
at six center-of-mass energies between 4.008 and 4.600 GeV, we observe the
processes $e^+e^-\rightarrow \phi\phi\omega$ and $e^+e^-\rightarrow
\phi\phi\phi$. The Born cross sections are measured and the ratio of the cross
sections $\sigma(e^+e^-\rightarrow \phi\phi\omega)/\sigma(e^+e^-\rightarrow
\phi\phi\phi)$ is estimated to be $1.75\pm0.22\pm0.19$ averaged over six energy
points, where the first uncertainty is statistical and the second is
systematic. The results represent first measurements of these interactions.
