Given a region $R$, let circle $C_1$ of radius $r_1$ be the largest circle contained in $R$, and let circle $C_2$ of radius $r_2$ be the smallest circle that contains $R$. The *fatness* of $R$ is given by $\frac{r_2}{r_1}$, and we say that $R$ is $\beta$-fat, for $\beta = \frac{r_2}{r_1}$. 

For simplicity, let $r_1=1$, and let $2$ be the side length of a square $S$. 

**I want to know the maximum number of $\frac{r_2}{r_1}$-fat pairwise-disjoint regions *with diameter at least 1* which can intersect $S$.** 

For example, if $r_2 = r_1=1$, then the regions are circles, and there are at most 6 circles with diameter at least 1 which can overlap $S$ without overlapping each other.
In the figure below, we have depicted a square and 7 circles, but the circles above and below the square cannot overlap the square without also overlapping the circle in the center of the square.

![overlapping circles][1]


  [1]: https://i.sstatic.net/B6YDm.png