# Partite Tur\'an-densities for complete $r-$uniform hypergraphs on $r+1$ vertices

## TL;DR

This paper establishes optimal density conditions for embedding complete r-uniform hypergraphs on r+1 vertices within (r+1)-partite hypergraphs, revealing linear thresholds and differences from the graph case.

## Contribution

It provides the first optimal density thresholds for hypergraph Turán problems on (r+1)-partite structures, extending classical graph results to hypergraphs.

## Key findings

- Optimal density condition in terms of induced subgraph densities
- Linear threshold functions for hypergraph embedding
- Difference in thresholds between graphs and hypergraphs

## Abstract

In this paper we investigate density conditions for finding a complete $r$-uniform hypergraph $K_{r+1}^{(r)}$ on $r+1$ vertices in an $(r+1)$-partite $r$-uniform hypergraph $G$. First we prove an optimal condition in terms of the densities of the $(r+1)$ induced $r$-partite subgraphs of $G$. Second, we prove a version of this result where we assume that $r$-tuples of vertices in $G$ have their neighbours evenly distributed in $G$. Third, we also prove a counting result for the minimum number of copies of $K_{r+1}^{(r)}$ when $G$ satisfies our density bound, and present some open problems. A striking difference between the graph, $r=2$, and the hypergraph, $ r \geq 3 $, cases is that in the first case both the existence threshold and the counting function are non-linear in the involved densities, whereas for hypergraphs they are given by a linear function. Also, the smallest density of the $r$-partite parts needed to ensure the existence of a complete $r$-graph with $(r+1)$ vertices is equal to the golden ratio $\tau=0.618\ldots$ for $r=2$, while it is $\frac{r}{r+1}$for $r\geq3$.