Problem #794: Is it true that every $3$-uniform hypergraph on $3n$...

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Progress

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.

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).

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 $\{1,2,3\}$.