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

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 $K^3_4$.