Book free $3$-Uniform Hypergraphs

TL;DR

This paper establishes an upper bound on the number of hyperedges in large 3-uniform hypergraphs that do not contain a specific structure called a $k$-book, extending understanding of hypergraph extremal problems.

Contribution

It proves a tight asymptotic upper bound on the maximum number of hyperedges in $k$-book-free 3-uniform hypergraphs, a new result in hypergraph extremal theory.

Findings

Maximum hyperedges is at most n^2/8 (1+o(1)) for large n.

Introduces bounds for $k$-book-free 3-uniform hypergraphs.

Advances extremal combinatorics in hypergraph structures.

Abstract

A $k$-book in a hypergraph consists of $k$ Berge triangles sharing a common edge. In this paper we prove that the number of the hyperedges in a $k$-book-free 3-uniform hypergraph on $n$ vertices is at most $\frac{n^2}{8}(1+o(1))$.