Population of P people, where each person knows K others, how many people mutually know each other If you have a population of $P$ people, where each person knows $K$ others within the population, (does not have to be mutual, i.e. if I know you, you don't necessarily know me), and $1 < K < P$, How many people at the least must mutually know each other? Is there a general formula for this minimum in terms of $P$ and $K?$

Here's an elementary start on investigating this interesting problem

Let $M$ be the minimum number of people in a clique. Then $M$ is non-zero if and only if $K>\frac{P-1}{2}.$

If $M=0$ then there is no pair such that each knows the other. The number of 'knowings', $KP$ , is therefore no greater than $\begin{pmatrix}P\\2\\\end{pmatrix}$ and so $K\le \frac{p-1}{2}.$
Conversely, suppose $K\le \frac{P-1}{2}.$ Consider the population arranged in a circle with everyone knowing the next $K$ people in the circle. Then no pair know each other.
