The Tur\'an number of Berge-K_4 in triple systems

TL;DR

This paper determines the maximum size of a triple system on n points that avoids Berge-$K_4$ configurations, showing that the extremal structure is a balanced complete 3-partite triple system.

Contribution

It establishes the Turán number for Berge-$K_4$ in triple systems and characterizes the extremal configuration as a balanced complete 3-partite system.

Findings

Maximum number of triples in a Berge-$K_4$-free system is achieved by a balanced 3-partite structure.

The extremal triple system is explicitly constructed as a complete 3-partite system.

The result extends Turán-type problems to Berge hypergraph configurations.

Abstract

A Berge-$K_4$ in a triple system is a configuration with four vertices $v_1,v_2,v_3,v_4$ and six distinct triples $\{e_{ij}: 1\le i< j \le 4\}$ such that $\{v_i,v_j\}\subset e_{ij}$ for every $1\le i<j\le 4$. We denote by $\cal{B}$ the set of Berge-$K_4$ configurations. A triple system is $\cal{B}$-free if it does not contain any member of $\cal{B}$. We prove that the maximum number of triples in a $\cal{B}$-free triple system on $n\ge 6$ points is obtained by the balanced complete $3$-partite triple system: all triples $\{abc: a\in A, b\in B, c\in C\}$ where $A,B,C$ is a partition of $n$ points with $$\left\lfloor{n\over 3}\right\rfloor=|A|\le |B|\le |C|=\left\lceil{n\over 3}\right\rceil.$$