Theorem 6.5.2 (Odd Oddtown).label Let $\cali{F}\suf\cali{P}([n])$ be a family such that for all $A\neq B\in\cali{F}$
- (a)
$\abs{A}\congruent{0}{2}$
- (b)
$\abs{A\cap B}\congruent{1}{2}$
- (1)
$\abs{\F}\leq n+1$
- (2)
$\abs{\F}\leq n$
Proof. Using the same set up as Oddtown Theorem 6.5.1 over $\bb{F}_{2}$ we have the hypothesis is equivalent to
- (1)
$\indi{A}\cd \indi{A}\congruent{0}{2}$
- (2)
$\indi{A}\cd\indi{B}\congruent{1}{2}$
\begin{align*}\F'\define \curl{A'=A\cup \curl{n+1}:A\in\F}\end{align*}
$\square$