Question 4
A set of $1990$ persons is divided into non-intersecting subsets so that:
1. No one in a subset knows all the others in the subset.
2. Among any three persons in a subset, at least two do not know each other.
3. For any two persons in a subset who do not know each other, exactly one person in that subset knows both.
(a) Prove that within each subset, every person has the same number of acquaintances.
(b) Determine the maximum possible number of subsets.
Knowing is mutual; an acquaintance is someone who is known. Every person is assumed to know oneself.
1. No one in a subset knows all the others in the subset.
2. Among any three persons in a subset, at least two do not know each other.
3. For any two persons in a subset who do not know each other, exactly one person in that subset knows both.
(a) Prove that within each subset, every person has the same number of acquaintances.
(b) Determine the maximum possible number of subsets.
Knowing is mutual; an acquaintance is someone who is known. Every person is assumed to know oneself.