Proving a Bound for Oddtown-Eventown or Clubtown

Question

Suppose we have a town with a set of residents $V$, where $|V| = n$. The residents like forming clubs, and we have clubs $C_1,C_2,\ldots,C_m \subseteq V$. We are interested in the maximum number of clubs $m$ that can be formed. There are some rules regarding clubs:

1. No two clubs are identical: all clubs are distinct.
2. A club must have an even number of members. $\forall i, |C_i|$ is even.
3. Any pair of two clubs shares an even number of members: $\forall i \neq j, |C_i \cap C_j|$ is even.

It follows from rule 1 that $m \leq 2^n$, as the set of all possible clubs is simply the power set of $V$.

Then it follows from rule 2 that $m \leq 2^{n-1}$, as the rule eliminates all odd subsets of $P(V)$. We note as a lemma that for a finite set $A$, exactly half the subsets of $P(A)$ are of odd cardinality. (Proof.)

Finally, the third rule further narrows down the bound: it implies that $m \geq 2^{\lfloor \frac{n}{2} \rfloor}$. However, I don't really understand the reasoning behind this. Can anyone explain the intuition or how to prove it?

Answer 1

Let $W$ be another town, with $k$ members. In town $W$, we still must have that all clubs are distinct, but town $W$ is not cultishly obsessed with parity like town $V$, so in town $W$, there is no harm in having a club with an odd number of members, or with odd numbers of intersections. Thus, as you already observed, $W$ can have $2^k$ clubs.

If $V$ has $n = 2k$ people then they can make clubs meeting the rules in the following way: they will split up into $k$ pairs. Each pair will pick a distinct person in town $W$, and they will form clubs copying the clubs from $W$, with pairs replacing singletons. (More precisely, a pair in $V$ is a member of club $C$ of $V$ copying club $D$ of $W$ if and only if the citizen of $W$ whom that pair picked is in $D$).

If $V$ has $n = 2k + 1$, just assume that one person is a loner who will never join any clubs and proceed as before.

This gives the lower bound that you mentioned.