Is there a way to estimate the minimal number of pairwise comparisons required to place complete ordering over a set of items? I have 200 items that I need users to compare. If there are 2 items, A and B, then the comparison is A before B or B before A. Since there 200 items, this is $\binom{200}{2} = 19,900$ possible combinations. These combinations are uniformly and independently drawn and shown to the users to evaluate. There is a very good chance that some combinations will not ever be shown, and hence, we will never know the comparison between those combinations.
My question is this: is there a way to estimate the minimum/required number of pairwise comparisons needed to place complete order over a set of items?
Example: We have 3 items {A, B, C}. If we compare {A,B} with B < A and {B,C} with B > C, then we can say the order is C, B, A.
 A: This is so far a very incomplete answer. I suppose 200 items, which we sample independently and uniformly.  If there is any prior information on the ordering, one could maybe do better than that.  I will represent the items as the integers $1,2,\dotsc, 200$ and suppose (in the simulations shown) that the true ordering (for one person)  is the usual ordering of integers. Then I suppose that the response on the pairs given is without error. We will need later to pass by such strong assumptions. So the question is: "how many pairs do we have to sample to get to know the true ordering?" 
I will answer a some what different question.  We will represent the sampled pairs as a directed graph, calculate the connected components of that graph and find the length of the longest connected component.  First I show a plot of a such graph, made with the R igraph package.  

This graph is simulated as above with 50 vertices and 50 pair comparisons, its largest component has 42 vertices and there is 8 isolated vertices. Looking closer at the largest component:

From this graph we can see that, for instance, both 17 and 21 is between 6 and 39, but there is no comparison between 17 and 21 themselves. So from such a graph we can calculate some fuzzy ranking, but it should be clear that if you want a complete ranking, the number of pair comparisons required will be very large!
To advance more I guess we will need some probability model on rankings, I will try to come back with that.
