@PhilHarmsworth Would you have a pdf or image of that article available? Very interested to read, but cannot locate.

If it follows from Besicovitch (eg. arxiv.org/pdf/2010.10040) then i dont see how, especially if $U$ is an open subset of small volume (possibly countably infinite genus in two dimensions).

@JensReinhold The Heisenberg manifold (compact three dimensional nilmanifold) does not satisfy your binomial coefficient bound. Compare Pete L. Clark's answer math.stackexchange.com/questions/434384/the-heisenberg-manifold. More generally, you should compare your binomial coefficient bound with the Toral Rank Conjecture. Manifolds with $M^n$ with toral rank $<n$ are likely going to be counterexamples to your $nCj$ lower bound.

Preferably any proof of Sponge Problem would make no appeals to width-volume inequalities (since Guth's motivation for Sponge Problem was to find new independant proof of Gromov's width-volume inequalities). I should have made this clear in the question. Do you know of elementary proof that $U$ can be partitioned into relative cycles of diameter $<100 \sqrt{\epsilon}$ (or $100^{100^{100}}\sqrt{\epsilon}$) in dimension two?

Evenmore explicit geometric $BG$ models from geometric topology (arithmetic, knots, surface groups, etc) are typically neither minimal nor simplicial. An exception is the well-rounded retract model $W=W_n$ of $PGL(\mathbb{Z}^n)\backslash PGL(\mathbb{R}^n)/SO(n)$, consisting of all flat $n$-dimensional tori whose $1$-systoles generate $\mathbb{Q}^n$ in rational homology. Here the cells are naturally parameterized by certain collections of simple closed curves on the torus. But the $G$-orbits of these collections is equivalent to the explicit $\mathbb{Z}G$-module structure of $\mathbb{D}$.

$G=\mathbb{Z}^n$ is exceptional in that the minimal $BG$ model is a compact orientable manifold, and therefore satisfies Poincare duality, Morse inequalities, etc., and in which Lusternick-Schnirelmann category is well studied. But in most cases, especially if $G$ is a Bieri-Eckmann duality group with dualizing $\mathbb{Z}G$-module $\mathbb{D}$ concentrated at a homological dimension $\nu>0$, then i don't think anybody knows how to count cells of any $BG$ model, nevermind the minimal $BG$ models which are no longer manifolds but necessarily singular topological objects.

@ArseniyAkopyan No, I don't know how. But would such an open set be any different from the disk? (If I connected them by roads, my roads would fill the disk). Do you have an image/picture of what your roads would look like?

Reportedly there exist "rigid/jammed" packings of arbitrarily low density, but the articles are behind degruyter paywall. degruyter.com/view/journals/zkri/221/5-7/…. If these novel packings N are *incompressible" then maybe $\epsilon^*$ is zero.

@BalarkaSen Yes!, i think your comment contains a nice improvement of $\epsilon^*$. If we replace any open set $U$ with an Apollonian packing $A$ of itself of minimal volume, then $vol(A)<vol(U)$ and $U$ e-embeds into $D$ iff $A$ e-embeds into $D$. So replacing the two kissing disks with an apollonian packing of minimal volume would improve $\epsilon^*$ by some factor.

Annuli appear to be no different than rectangles (with respect to e-embeddings), but I don't have a precise criterion for which rectangles e-embed into $D$ (except volume and disjoint disks). All this sponge stuff appears to have begun with Ya. Barzdin and A. Kolmogorov. On realization of nets in 3-dimensional space. Problems of Cybernetics, 19:261–268, 1967. But I have not studied that paper. Some introduction can be found in last chapter of P.G.Adey's thesis pgadey.com/ut-thesis.pdf .

If reference disk $D$ has radius $r=1$, then the kissing disks have radius $r_1=r_2=1/2$. The interiors of the kissing disks are disjoint, and can be embedded into $D$, but every arbitrarily small $\epsilon$-thickening of $D_1 \cup D_2$, or pair of kissing disks of radius $r_1=r_2=1/2+\epsilon$ cannot be expanded embedded into $D$.