Q:

Photo Booth problem

There are $n$ people.
There is a Photo Booth in which they can enter at most $m$ people at one time.
They want to get a picture with all other person together.
Please solve the $F(n,m)$; minimum number of times to get required pictures.
For example, there are people: $A$, $B$, $C$ and $D$, and a Photo Booth in which they can enter at most $3$ people at one time. When they form a group $\left\{ A, B, C \right\}$, $\left\{B, C, D \right\}$ and $\left\{A, C, D\right\}$, $A$ can get pictures with $B$, $C$ and $D$ ,etc. In this case, $F(4,3)=3$
See also Question of difficult matrix problem, minimum number of times

A:

A $(v,k,t)$ covering is a family of subsets of $\{1,2,3\dots v\}$ such that each subset has $k$ elements and every subset of $\{1,2,3\dots,v\}$ that has exactly $t$ elements is contained in one of the subsets of the family.
What you want to find is the minimal subsets that can be in a $\{v,k,2\}$ covering. Here is a repository of such coverings.

