Let
A collection
of subsets
is said to be separating, if for every pair
there is a set
so that
contains just one element.
is said to be covering, if every element of
is contained in at least one set
What is the smallest value
of
so there is a set
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?
Školjka