Turán numbers and switching

Using a switching operation on tournaments we obtain some new lower bounds on the Turán 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.