An exact Tur\'an result for tripartite 3-graphs

TL;DR

This paper establishes an exact Turán-type theorem for 3-uniform hypergraphs, identifying the maximum size and structure of hypergraphs avoiding specific subgraphs, extending classical results from graph theory to hypergraphs.

Contribution

It proves the unique extremal structure for 3-graphs avoiding certain subgraphs, generalizing Mantel's theorem to tripartite 3-graphs.

Findings

The balanced complete tripartite 3-graph maximizes size for certain forbidden subgraphs.

Unique extremal 3-graph structure identified for all but one small case.

Extension of Bollobás's classical result to more complex hypergraph configurations.

Abstract

Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let $K_4^-=\{123,124,134\}$, $F_6=\{123,124,345,156\}$ and $\mathcal{F}=\{K_4^-,F_6\}$: for $n\neq 5$ the unique $\mathcal{F}$-free 3-graph of order $n$ and maximum size is the balanced complete tripartite 3-graph $S_3(n)$ (for $n=5$ it is $C_5^{(3)}=\{123,234,345,145,125\}$). This extends an old result of Bollob\'as that $S_3(n) $ is the unique 3-graph of maximum size with no copy of $K_4^-=\{123,124,134\}$ or $F_5=\{123,124,345\}$.