TITLE: Kissing number and overlapping number
QUESTION [7 upvotes]: Let $S$ be a certain family of geometric objects (e.g, the family of unit squares).
The kissing number of $S$ is the maximum number of nonoverlapping elements of  $S$ that can touch one element of $S$. See here an example where $S$ is the family of unit-squares: The kissing number of a square, cube, hypercube? 
Define the overlapping number of $S$ as the the maximum number of nonoverlapping elements of  $S$ that can overlap one element of $S$. 
Obviously, the overlapping number is not larger than the kissing number since, if the central object has to be overlapped (not only touched), there is less room for the peripheral objects.
What else is known about the relation between the kissing number and the "overlapping number"? In particular, is there any bound on the ratio: kissing-number/overlapping-number?
EXAMPLE: When $S$ is the family of axis-parallel unit-squares, the overlapping number is 4 and the kissing number is 8. So the ratio is 2.

REPLY [7 votes]: Even for a set almost the same as the one you mention in your question, unit squares but with the four corners removed, the overlapping number is larger than the kissing number. Only six cornerless squares can kiss another cornerless square, because kissing can only happen along the sides of a square, and if the kissed square has two kissing squares on one of its sides then it must have only one each on the two adjacent sides. However, seven nonoverlapping cornerless squares can overlap a central cornerless square: