Sorry that was just thousands. An interesting consequence of this actually, is that an algorithm that is online for components is highly preferable (as it would prevent us from needing to choose a value for $l$, we could instead make some choices around convergence [i.e. we've added 10 new dimensions and $V^*$ has remained stable)

Our practical values are; $10^5 < |V| < 10^7$, $n \approx 100$, $k < 10$... $l$ is a bit more tricky. $l$ actually represents samples from a "dataset distribution", so we should expect that as $l$ grows, $V^*$ converges to some stable set. It's hard to know when this will happen, and it will likely depend on $|V|$ and $|k|$, but my guess would be $O(1000s)$