User blog comment:Wythagoras/Graphs/@comment-11227630-20140913013819

I'm not sure about the strength of my cycle level ordering - maybe it's below \(\vartheta(\Omega_\omega)\). According to this, we don't know whether your "K3,3 with two vertices at one of the ends" has level \(\theta(\Omega_\omega,1)\), if it works in that way.