  In this note, we give short proofs of two theorems in extremal set theory.
The first one is a determination of the maximum size of a nontrivial
$k$-uniform, $d$-wise intersecting family for $n\ge
\left(1+\frac{d}{2}\right)(k-d+2)$, which improves upon a recent result of
O'Neill and Verstra\"{e}te. Our proof also extends to $d$-wise,
$t$-intersecting families, and from this result we obtain a version of the
Erd\H{o}s-Ko-Rado theorem for $d$-wise, $t$-intersecting families.
  The second result partially proves a conjecture of Frankl and Tokushige about
$k$-uniform families with restricted pairwise intersection sizes.
