My motivation is to apply some graph theoretic methods in real analysis (like in a simple proof that every open real set $ A $ is coutable union of disjoint graph - take the connected component of the intersection graph of $ A = \bigcup_\alpha I_\alpha$: $G=(A,E) $ where $ xEy $ iff there exists an interval $ I_\alpha $ such that $ x,y \in I $. More proofs can be found in 1):

Below is a statement of Berkuwich covering theorem (A statement of [2] with few minor changes):

Let $ N $ be a positive integer. There is a constant $ K = K(2) $ with the following property. Let $ B = \{ B_\alpha \}_\alpha$ be any collection of balls in $ \mathbb{R}^2 $ with the property that the interior of no ball contains the center of any other. Then we may write $ B = \mathcal{B}_1 \cup \ldots \cup \mathcal{B}_k $ so that each $ \mathcal{B}_j $ $ j = 1,\ldots,K $ is a collection of balls with pairwise disjoint closures.

Here by a ball we mean a set $ B $ satisfying $ \mathbb{B(x,r)} \subseteq B \subseteq \overline{\mathbb{B}}(x,r) $ for some $ x \in \mathbb{R^2} $ and some $ r> 0 $.

Than apply that idea we can define the **Sphere of Influence Graph** of the collection to be the graph $ G = (B,E) $ such that two intersect balls connected by an edge.

This idea discussed in [3][4].

This SIG graph has also the property that the interior of no ball contains the center of the other.

In that setting we need to find the chromatic number of the SIG of $ B $.

Firstly, this leads to an (known) obseration that we need only find a bound for finite $ B $ (using De Bruijn–Erdős theorem). So we can assume that B is finite.

Than, one can conjeture that under the constain we get a planar graph, sadly it's not the case, but we may consider their cross-number.

However, even if we have their cross-number what can we get about their chromatic number?

So there are two questions:

**What is the cross-number of the SIG of $ B $ under the constrain that the interior of no ball contains the center of any other.****A good approximation is useful too.****Even we find the cross-number, what can we deduce about their chromatic number?**Maybe it's possible to apply some computational methods (like in the four-color theorem proof).**A good approximation is useful too.**

I know that this strateghy isn't the best one to find the Berkuwich number (works only at the plane), but I find it interesting because of the graph-theoretic asspects. Thanks.

Remark:

The question arises an Eudclid-geometry question: **Is there exists a constant $ k $ such that if we take any collection $ \{ C_i \}_{i=1}^k $ of closed ball that the interior of no ball contains the center of any other, and than draw a segment between centers of intersected circle, than the maximal number of segment intersections is at most $ k $? What is the value of that number?**

1 Orest Xherija (https://math.stackexchange.com/users/56880/orest-xherija), Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs], URL (version: 2015-04-18): https://math.stackexchange.com/q/318299.

[2] Krantz, Steven G., and Harold R. Parks. Geometric integration theory. Springer Science & Business Media, 2008.

[3] Soss, Michael A. "On the size of the euclidean sphere of influence graph." CCG. 1999.

[4] Erdös, Paul, et al. "Intersection graphs for families of balls in RN." European Journal of Combinatorics 9.5 (1988): 501-505.