Let
![N = \{1,2 \ldots, n\}, n \geq 2.](/media/m/1/0/7/107c5997bc1d084be4cec7d70bb23434.png)
A collection
![F = \{A_1, \ldots, A_t\}](/media/m/3/c/8/3c8cec75f02412202460da4830d27294.png)
of subsets
![i = 1, \ldots, t,](/media/m/a/1/f/a1f6f06ce683083e29d43be8204ffead.png)
is said to be separating, if for every pair
![\{x,y\} \subseteq N,](/media/m/7/c/6/7c6c033d0f8c5a557a80c84b15d4a0bb.png)
there is a set
![A_i \in F](/media/m/7/9/6/79656d47678c169060783847c6fc0324.png)
so that
![A_i \cap \{x,y\}](/media/m/7/c/1/7c1ac13cff18811f90b212cdc48f1309.png)
contains just one element.
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
is said to be covering, if every element of
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
is contained in at least one set
![A_i \in F.](/media/m/f/6/6/f66858c768c14345f27492c6b2ce3a3b.png)
What is the smallest value
![f(n)](/media/m/d/3/e/d3e47283bffbbf24c97f0c6474d5a82d.png)
of
![t,](/media/m/6/5/2/6529c7beabef0ae40757d888c6618632.png)
so there is a set
![F = \{A_1, \ldots, A_t\}](/media/m/3/c/8/3c8cec75f02412202460da4830d27294.png)
which is simultaneously separating and covering?
%V0
Let $N = \{1,2 \ldots, n\}, n \geq 2.$ A collection $F = \{A_1, \ldots, A_t\}$ of subsets $A_i \subseteq N,$ $i = 1, \ldots, t,$ is said to be separating, if for every pair $\{x,y\} \subseteq N,$ there is a set $A_i \in F$ so that $A_i \cap \{x,y\}$ contains just one element. $F$ is said to be covering, if every element of $N$ is contained in at least one set $A_i \in F.$ What is the smallest value $f(n)$ of $t,$ so there is a set $F = \{A_1, \ldots, A_t\}$ which is simultaneously separating and covering?