Assume that such a board cannot be constructed. Take some member of the society A. Either there is some person B whose list does not share any candidates with person A's list, or everyone's list shares candidates with person A's list. In the latter case, we can simply make the board of governors the 10 people on person A's list. Thus, there is some person B whose list does not share any candidates with person A's list. Now, randomly split person A's list into two smaller lists of 5 candidates, A1 and A2. Do the same for person B, split his list into two smaller list of 5 candidates, B1 and B2. Choosing Ai and Bj as the board must make someone unhappy, call this person Cij. Now consider the six people A,B,C11,C12,C21,C22. We can choose two candidates X and Y such that at least one of them appears on everyones list. Either X or Y has to appear on A's list, and either X or Y has to appear on B's list. Because their lists are mutually disjoint, one of X/Y appears on A's list, and one of X/Y appears on B's list. WLOG, X is on A's list and Y is on B's list. Then, X is either in A1 or A2 (WLOG, A1.) and similarly Y is WLOG in B1. Then, neither X nor Y can be on C11's list, or someone from A1 and B1 would be on C11's list and choosing A1 and B1 as the board would make C11 happy, a contradiction.