Combinatorics

Given the range 1-10
Form all possible pairs: [1,2], [1,3], [1,4], [1,5], [2,3], [2,4],[2,5],[3,4],[3,5],[4,5]

I would like some help with an algorithm to do the following:

Combine numbers into a minimal list of five numbers to ensure all pairs are covered with minimum duplication.
In the list below all pairs are covered i.e 1 with every other number, 2 with every other number etc.
For example the last line could be removed because the second and second last line already have these pairs represented.


1,2,3,4,5
1,6,7,8,9
1,10,2,6,7
2,8,9,10,3
3,6,7,8,9
3,10,4,6,7
4,8,9,10,5
5,10,6,7,8