6 problems

Sós's conjecture. The unique extremal hypergraph for the Fano plane is $B_n$.

Let $\textbf{F}$ be the Fano plane, the $3$-graph on seven vertices with edges $\{axx',ayy',azz',xyz',xy'z,x'yz,x'y'z'\}$. Let $B(n)$ be the balanced complete bipartite $3$-graph o…

Let $G$ be the iterated blow-up of the complement of the Fano plane. It is $J_4$-free and has codegree squared density at least $0.28$. The complement-of-Fano blow-up conjecture fo…

Let $\mathbb{F}$ be the Fano plane, and let $B_n$ be the complete balanced bipartite $3$-graph on $n$ vertices. For a $3$-graph $H$, write $\operatorname{co}_2(H)$ for its codegree…

Let $H$) be a $3$-graph, and let $\mathrm{ex}_2(n,H)$ denote the maximum minimum co-degree of an $n$-vertex $3$-graph containing no copy of $H$. Let $\textbf{F}$ be the Fano plane,…

Let $PG_2(2)$ denote the Fano plane, viewed as a $3$-uniform hypergraph, and let $\operatorname{ex}_2(n,PG_2(2))$ be its exact codegree threshold on $n$ vertices. Mubayi's conjectu…