The 3-uniform hypergraph on Fin 9 given by Harris: the 27 triples containing exactly one vertex from each of the three parts {0, 1, 2}, {3, 4, 5}, {6, 7, 8}, together with the triple {0, 1, 2}. A triple is a transversal of the three parts exactly when the three values of v ↦ v / 3 it produces are distinct.

Is it true that every $3$-uniform hypergraph on $3n$ vertices with at least $n^3+1$ edges must contain either a subgraph on $4$ vertices with $3$ edges or a subgraph on $5$ vertices with $7$ edges?

Harris has provided the following simple counterexample to the problem as stated: the $3$-uniform graph on ${1,\ldots,9}$ with $28$ edges, formed by taking $27$ edges by choosing one element each from ${1,2,3},{4,5,6},{7,8,9}$, and then adding the edge ${1,2,3}$.

Balogh has observed that this problem is probably misstated by Erdős - indeed, every graph with $5$ vertices spanning $7$ edges contains a graph on $4$ vertices spanning $3$ edges, so the second condition can be dropped.

Harris has provided the following simple counterexample to the problem as stated: the $3$-uniform graph on ${1,\ldots,9}$ with $28$ edges, formed by taking $27$ edges by choosing one element each from ${1,2,3},{4,5,6},{7,8,9}$, and then adding the edge ${1,2,3}$.

This problem is then now asking how many edges a $3$-uniform hypergraph can have before it contains $K_4$ minus an edge, and whether the critical edge density is $2/9$. In fact there is a construction of Frankl and Füredi [FrFu84] showing it must be at least $2/7$, which is the conjectured truth (although Turán conjectured before [Er69] that the edge density was $1/4$, and so likely there is simply a typo in this problem's statement).