Limits on symmetric monitoring

Let n be the number of people that a person can maximally deal with. Then the set of people who are organised in groups of people who monitor each other, that is monitor each other mutually within the same group and mutually as groups over different groups, can not exceed (n+1)² in size.

Proof. Let m be the number of people in some group. Then the internal monitoring leaves the group with a maximum of m(n-(m-1)) members of other groups it can still monitor, i.e. m_i(n-(m_i-1)) is for every i an upper bound for the sum of all m_j, j≠i.

Considering the smallest m, we find that it can not be smaller than the sum of the other m divided by n and in particular that there can not be more than n+1 groups.

And considering the biggest m, we find that it can not exceed n+1.

Actually, Σ_jm_j=m_i(n+2-m_i) for every i, is being solved by

m_i=n/2+1, where i=1, ..., n/2+1,

which is a good candidate for the maximal sum. In general the solution to m_i(n+2-m_i)=m_j(n+2-m_j) for some pair i≠j, 0<m_i<n+2, is

n/2+1±((n/2+1)²-m_i(n+2-m_i))^0.5=n/2+1±(m_i²-(n+2)m_i+(n/2+1)²)^0.5=n/2+1±(n/2+1-m_i),

that is all m_i must take one of two equidistant values to n/2+1. Let m and m' be those two values, i.e. m'=n+2-m and m=n+2-m'. Let a be the number of groups of size m and b be the number of groups of size m'. Then am+bm'=mm' and this value has to be maximal. But m(n+2-m) is maximal for m=n/2+1.

Hence the actual limit to the size of the set of people who monitor each other should be (n/2+1)², but my point here has little to do with the exact value. It is this:

The monitoring of a potentially infinite set of people must be structured and effective structures are at least implicitly hierarchical, in so far that they have a center.