Two questions of Erd\H{o}s on hypergraphs above the Tur{\'a}n threshold}

TL;DR

This paper investigates Erdős's questions on hypergraph properties above the Turán threshold, showing negative results for certain complete 3-uniform hypergraphs and their subgraphs, especially for small and large vertex counts.

Contribution

It provides counterexamples and density comparisons that answer Erdős's questions negatively for specific hypergraph configurations.

Findings

Negative answer for s=4 with small n

Negative answer for s=4 for all large n

Turán density of ${K^3_5}^-$ exceeds that of $K^3_4$

Abstract

For ordinary graphs it is known that any graph $G$ with more edges than the Tur{\'a}n number of $K_s$ must contain several copies of $K_s$, and a copy of $K_{s+1}^-$, the complete graph on $s+1$ vertices with one missing edge. Erd\H{o}s asked if the same result is true for $K^3_s$, the complete 3-uniform hypergraph on $s$ vertices. In this note we show that for small values of $n$, the number of vertices in $G$, the answer is negative for $s=4$. For the second property, that of containing a ${K^3_{s+1}}^-$, we show that for $s=4$ the answer is negative for all large $n$ as well, by proving that the Tur{\'a}n density of ${K^3_5}^-$ is greater than that of $K^3_4$.