Tur\'{a}n numbers and switching

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.