An exact extremal result for tournaments and 4-uniform hypergraphs

TL;DR

This paper determines the maximum number of hyperedges in certain 4-uniform hypergraphs with specific vertex set conditions, extending previous results by leveraging skew-symmetric conference matrices.

Contribution

The paper provides an exact extremal result for 4-uniform hypergraphs using skew-symmetric conference matrices, covering cases not previously solved.

Findings

Solved the problem for n ≡ 0 mod 4 and n ≡ 3 mod 4

Extended known results from r=3 to r=4

Relied on the existence of skew-symmetric conference matrices

Abstract

In this paper, we address the following problem due to Frankl and F\"uredi (1984). What is the maximum number of hyperedges in an $r$-uniform hypergraph with $n$ vertices, such that every set of $r+1$ vertices contains $0$ or exactly $2$ hyperedges? They solved this problem for $r=3$. For $r=4$, a partial solution is given by Gunderson and Semeraro (2017) when $n=q+1$ for some prime power number $q\equiv3\pmod{4} $. Assuming the existence of skew-symmetric conference matrices for every order divisible by $4$, we give a solution for $n\equiv0\pmod{4} $ and for $n\equiv3\pmod{4}$.