Let us fix an alphabet $\Sigma=\{0,1,2,\dots,n-1\}$ for a given $n$ and consider words over $\Sigma$, i.e. elements of $\Sigma^*$.
Consider the following 5 words:
$000\\001\\002\\010\\100$
They satisfy the property that for every two indices $1\le i_1,i_2\le 3$, if we take the subwords obtained from the letters in the indices $i_1, i_2$, we will end up with two identical subwords. For example, if $i_1=1,i_2=3$ then both $000$ and $010$ give the subword $00$.
What I want is this: to find a value $t$ such that for all the possible sets of 5 distinct words of equal ($\ge t$) size, there exists some choice of $t$ indices which gives rise to distinct subwords for all the words. My example shows that $t=2$ will not work.
I only need a solution for the case $n=5$ right now, but I'm also interested in the most general case, where we need to find a $t$ that is good not only for sets of size $5$ but for sets of size $r$ for some integer $r$. Hence, I'm actually asking about computing/bounding a function $t(n,r)$ which depends on both the size of the alphabet and the cardinality of the set of words.