Tournaments, 4-uniform hypergraphs, and an exact extremal result

TL;DR

This paper constructs an infinite family of 4-uniform hypergraphs with maximum edges under specific vertex subset conditions, using quadratic residues and Paley tournaments, connecting to prior asymptotic results.

Contribution

It introduces a new explicit construction of extremal hypergraphs using quadratic residues and Paley tournaments, linking combinatorial design with algebraic structures.

Findings

Constructed infinite family of extremal hypergraphs

Connected explicit construction to Baber's asymptotic results

Analyzed a switching operation preserving hypergraph properties

Abstract

We consider $4$-uniform hypergraphs with the maximum number of hyperedges subject to the condition that every set of $5$ vertices spans either $0$ or exactly $2$ hyperedges and give a construction, using quadratic residues, for an infinite family of such hypergraphs with the maximum number of hyperedges. Baber has previously given an asymptotically best-possible result using random tournaments. We give a connection between Baber's result and our construction via Paley tournaments and investigate a `switching' operation on tournaments that preserves hypergraphs arising from this construction.