Tur\'{a}n numbers and switching

TL;DR

This paper introduces a switching operation on tournaments to derive new lower bounds on Turán numbers for specific hypergraphs, and explores the uniqueness and structure of extremal examples using Fourier analysis and generalizations of classical results.

Contribution

It develops a switching technique for tournaments, establishes the uniqueness of extremal Paley tournament examples, and generalizes the concept to higher-order tournaments with a new formula for switching classes.

Findings

New lower bounds on Turán numbers for 3-uniform hypergraphs.

Uniqueness of extremal Paley tournament examples proven via Fourier analysis.

Generalization of switching operations to higher-order tournaments with a formula for switching classes.

Abstract

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.