8 problems

Let $F$ be a $(k-1)$-graph, and let $\mathcal{S}_F$ be the $k$-graph obtained by adjoining an apex vertex $v$ and including $v\cup e$ as an edge for every $e\in E(F)$. For…

Nagle's conjecture.

The codegree-squared density conjecture for $K_5^4$. The lower bound is tight:

The lower-bound sharpness conjecture for $K_5^<$. All four lower bounds are sharp:

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 $G$ be the iterated blow-up of an edge, obtained from a complete balanced $3$-partite $3$-graph by inserting a complete balanced $3$-partite $3$-graph recursively in each part.…

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…

Bohman–Frieze–Mubayi–Pikhurko's conjecture. The balanced construction $T_{A,B,C}$ is asymptotically optimal for the codegree problem for $F_{3,2}$; equivalently,