
a combinatorics problem
I have B different labels and C different objects. Each label can be assigned to more than one object at the same time, and each object can have more than one label.
I want to assign every label to exactly X of my C objects, while ensuring that no object has more than Y different labels.
For some values of C, this will not be possible. My question is: What is the smallest value of C for which I can do this assignment?

my solution
...is in the next post....

my solution
I think the lowest number of objects is given by
$\displaystyle X$ if $\displaystyle B \leq Y$
and $\displaystyle X+(BY)\left\lceil X/Y \right\rceil $ otherwise
where with $\displaystyle B \leq Y$ we have every object getting all labels, and so X objects are required, while otherwise $\displaystyle \left\lceil X/Y \right\rceil$ extra objects are required to be assigned the labels not used up by the first X objects.