Krdytkyu

For $ninBbb N$, consider $n$ points $x_1,ldots, x_n$ in the unit square $Q=[0,1]^2$. Let $f(x_1,ldots, x_n)$ denote the minimal total edge length of a tree with $x_1,ldots, x_n$ as vertices.
Let
$$ L_n=sup_{x_1,ldots, x_nin Q}f(x_1,ldots,x_n).$$
Then in Prove that there exists a graph with these points such that G is connected, it was asked to show that $$tag1forall nge 3colon L_nle csqrt n $$
is true for $c=10$ and false for $c=1$.

In my answer (see there), an inductive argument based on subdividing the square showed that $(1)$ is true for $c=2sqrt 2approx 2.828$, and has infinitely many counterexamples for $c=1$.

In a later addendum to my answer, I showed that
$$ tag2exists ell_0colon forall ncolon L_n<ell_0+csqrt n$$
holds for any $c>frac 4{sqrt{2sqrt 3}}approx 2.149$ and dared to

Conjecture. $$inf{,cinBbb Rmid (2) text{ is true},}=frac{4}{sqrt{2sqrt 3}}. $$

I dared to do so because the constant stems from the factor $frac{pi}{2sqrt 3}$ known from the densest circle packing.

My question for you us: Is my conjecture true?

Silly me! After a good night's rest, this direction is of course the trivial part!

For $mgg 0$, the lattice $frac 1mBbb Z[frac{1+isqrt 3}2]$ has $frac {2m}{sqrt 3}+O(1)$ rows of $m+O(1)$ points each in the unit square, so that's $n=frac 2{sqrt 3}m² +O(m)$ points with minimal distance $frac 1m$ between them. Any tree with these vertices certainly has edge length
$$ell =frac{n-1}m=frac 2{sqrt 3}m+O(1) =frac{sqrt 2}{sqrt[4]3}sqrt n+O(1)=frac 2{sqrt{2sqrt 3}}sqrt n+O(1).$$
I suppose, a spurious factor of $2$ somehow slipped into the calculations leading to the conjecture.