Suppose I have a set of finite sets: $X = \{V_1, V_2, V_3\}$ where each set called $V_i$ in $X$ contains a number of symbols (i.e. $V_1 = \{a,b,c\}$). $Z$ contains all of the Cartesian products of pairs of sets chosen from $X$. So in this case $Z = \{V_1 \times V_2, V_1 \times V_3, V_2 \times V_3\}$. A "word" is a set with each character at position $i$ drawn from the corresponding $V_i$. The set of all words is $W = V_1 \times V_2 \times V_3$. A "sentence" is an ordered set of "words" taken from $W$.

How many "sentences" of minimal length can I make such that every pair from $Z$ is in that "sentence" at least once? Minimal means fewest possible number of "words."

single$V_i$ to be valid. For example if $V_1 = \\{a,b,c\\}$ and $V_2 = \\{1,2,3\\}$, "words" starting with $\\{a,1,...\\}$ would be valid but "words" like ${a,a,...}$ or ${1,1,...}$ would not be since a "word" is an ordered set where each character at position $i$ mayonlybe drawn from the corresponding $V_i$. – Misha Jun 16 '11 at 17:41