BraveDingo215

Wow, I love that this is interesting to people! Merged the PR.

Sorry, my previous comment was misleading on this point; 4/3 is conjectured, not proved (Conjecture 1.1 in the preprint): the supremum of critical constants over common-point cells equals 4/3, approached but not attained along the record ladder; the only proved lower bound is the k=17 certificate. Also, importantly, 4/3 is not an extrapolation from the ladder (Theorem 5.1).

The 3/2 you found is a proven theorem but says a more narrow thing: it is the codimension-two case, complement mass q=2, and the record cells are not there (k=17 has q=11), so it does not bound the ladder.

So yes, I should be clear: the gap is still [1.28249, 2], with 2 itself the conjecture for the universal constant (Conjecture 1.2). We conjecture that 4/3 is the limit for the common-point class, and large searches will find counterexamples in this class going to 4/3, but we don't have an explicit family of candidates approaching this limit. Again, sorry for the confusion, hope this clears things up.

Thanks for the quick turnaround! Just wanted to add what I forgot to mention in the description: the limit for this class of examples is $\tfrac 43$ as the number of terminals in the graph goes to $\infty$. So with a huge computation, we could probably get to $\tfrac 43 - \epsilon$; $1.28249...$ is a specific graph with $17$ terminals.