TITLE: Difficult variation of the committee problem
QUESTION [7 upvotes]: It is a trivial exercise in the pigeonhole principle to show that if an organization contains $m$ people and forms disjoint committees of $n$ members each, then at most
$$\bigg \lfloor \frac{m}{n} \bigg\rfloor$$
committees can be formed.
However, I have recently attempted a seemingly simple variation on this problem: what if the committees need not be disjoint, but no two committees can share more than one member.
It seems like another easy exercise in the pigeonhole principle, but I have been unable to come up with a formula for the largest possible number of committees in terms of $m$ and $n$. Does anyone know of such a formula?

REPLY [3 votes]: This is a well-known problem which I often meet in different forms. For given $m$ and $n$ let $c=c(m,n)$ be the largest possible number of committees. As far as I know, exact formulas for $c(m,n)$ are known only in particular cases, for small $m,n$ and two infinite series. There are ${m\choose 2}$ different pairs of organization members. On the other hand, each committee provides ${n\choose 2}$ such pairs, and no pair can be provided by two committees. This gives an upper bound $c\le\frac{m(m-1)}{n(n-1)}$. This upper bound is exact iff there exists a Steiner system $S(2,n,m)$. For particular values of $m,n$ their construction is based on finite projective planes. But such planes are over a finite field of a order $q$, which exists iff $q$ is a power of a prime, and there are no known other $q$ for which these exist a Steiner system $S(2, q, q^2)$ (or, equivalently,  $S(2,q+1,q^2+n+1)$). Nevertheless,  when $m$ is close to $n^2$ this approach provides a rather tight asymptotic lower bound for $c(m,n)$, because for sufficiently big $n$ there exists a prime number $n-n^{0.525}\le q\le n$ (see the paper “The difference between consecutive primes II” by
R. C. Baker, G. Harman, and J. Pintz). So estimating $c(m,n)$ for concrete $m$ and $n$ we usually pick a basic committee pattern provided by a finite projective plane and then try to improve the construction. The upper bound sometimes can be improved too by more subtle and complicated estimations. For instance, a current bounty question asks about minimum $m$ such that $c(m,10)\ge 40$. The current bounds are $74\le m\le 85$.