In this year's presidential election of Graphistan, 20 nominees compete for the position of head of state. It is your job to schedule a series of public debates between them on national television. There are some constraints you must consider:
1. We do not want it too crowdy, so at most 10 contenders may appear in the same TV show.
2. It is an accepted custom, that candidates shall not attack anyone who is absent. Therefore, ensure that every pair of candidates appears in at least one show together.
a) How can you schedule the events with the minimum number of shows?
b) What if there are 22 candidates and the limit is raised to 11 candidates per show?
(In reply to First thoughts
Part a) Each candidate must be on the debate stage with 19 other candidates, across all the debates. If a candidate was in only 2 debates, them he would be debating at most 18 other candidates. Therefore, each candidate must be in at least 3 debates. Therefore, a lower bound on the minimum number of debates is 6. (20*3/10). And Tomarken has demonstrated that 6 is possible.
Part b) By a parallel argument, 6 is a lower bound on the minimum number of debates. Not sure if it is achievable.