15 problems

Tree-suspension optimality conjecture. Then

Let $F$ be a $k$-graph. For integers $1\leq\ell\leq k-3$, let $\pi_{\rm u}^{(\ell)}(F)$ denote its $\ell$-uniform Turán density. An $(\ell+1)$-vanishing order is an ordering of…

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…

For integers $k\geq4$ and $\ell\geq3$ with $\ell<k$, let $F$ be a $k$-graph. An $\ell$-vanishing order of $F$ is an ordering of $V(F)$ admitting an $\ell$-vanishing coloring with t…

Let $f_r(n,v,e)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph such that every set of $v$ vertices spans fewer than $e$ edges. For fixed integers…

Higher-uniformity extremal construction conjecture. For all sufficiently long $L$ relatively prime to $r$, the extremal $C_L^{(r)}$-hom-free constructions on sufficiently large num…

A $3$-graph $F$ is layered if there is a function $f:V(F)\to\mathbb N$ such that: every edge has exactly one vertex whose label is strictly greater than the other two; edges with t…

Let $G^3_{n,p}$ be the random $3$-uniform hypergraph on $n$ vertices in which each $3$-edge is present independently with probability $p$, and let $C^3_4$ be the $3$-uniform linear…

Bollobás–Leader–Malvenuto conjecture. For all $t\geq 2$,

Optimal construction conjecture.

Partially directed triangle conjecture. For all sufficiently large $n$, every $n$-vertex partially directed $k$-graph with $\alpha\binom{n}{k}$ undirected edges and…

Let $r\geq 2$, and let $\mathcal{L}_{n,k}^{(r)}$ be the family of all tight linear forests of order $n$ with $k$ edges in $r$-graphs. Here a tight linear forest is an $r$-graph who…

Complete-host conjecture. If

Let $r\geq 4$ and let $H_{2r}^{M_2^r}$ be the extension of the $r$-uniform matching $M_2^r$ consisting of two pairwise disjoint edges. Let $G$ be an $r$-graph on $[n]$, and let…

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,