  Using a switching operation on tournaments we obtain some new lower bounds on
the Tur\'{a}n number of the $r$-graph on $r+1$ vertices with $3$ edges. For
$r=4$, extremal examples were constructed using Paley tournaments in previous
work. We show that these examples are unique (in a particular sense) using
Fourier analysis.
  A $3$-tournament is a `higher order' version of a tournament given by an
alternating function on triples of distinct vertices in a vertex set. We show
that $3$-tournaments also enjoy a switching operation and use this to give a
formula for the size of a switching class in terms of level permutations,
generalising a result of Babai--Cameron.
