30 problems

Let $G$ be an extremal $\ell$-uniform hypergraph for Problem 13 on $n$ vertices, and let $K_{t+1}^{\ell}$ denote the complete $\ell$-uniform hypergraph on $t+1$ vertices. An isolat…

Extremal characterization conjecture. $f_r(s)$ achieves the maximum for either $\mathcal{K}_{s+r-1}^r$, or the hypergraph $\mathcal{G}$ on $s+r$ vertices obtained by this minimum-e…

Let $M_s^r$ be a matching of size $s$ in an $r$-uniform hypergraph, and let $\operatorname{ex}_r(n,M_s^r)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergrap…

Tight tree Ramsey growth conjecture. For $r\geq 5$, if $T$ is a non-trivial tight $r$-tree, then there exist constants $c_1,c_2>0$ such that, for every positive integer $n$,

The family-level tower lower-bound conjecture. If $k>s$, then there exists a positive constant $c$ such that

The tower lower-bound conjecture. If $k>s$ and $H$ is an $s$-tightly connected $k$-graph that is not $k$-partite, then there exists a positive constant $c$ such that

The iterated-tripartite classification conjecture. For a 3-graph $H$, there exists a constant $c$ depending only on $H$ such that

Let $H$ be a connected $k$-uniform hypergraph, let $\mathcal{L}(H)$ be its Laplacian tensor, let $\mathcal{A}(H)$ be its adjacency tensor, and let $\rho$ be the spectral radius of…

Let $H$ be a connected $k$-uniform hypergraph, let $\mathcal{A}(H)$ be its adjacency tensor, let $\rho$ be its spectral radius, and let $\mathbb{V}_{\rho}(\mathcal{A}(H))$ be the pro…

Graft transformation conjecture. For $s\geq t\geq 2$,

Zheng's conjecture. For every $k$-uniform hypertree $T$ with $k\geq 3$, the multiplicity of the zero Laplacian eigenvalue is $k^{|E(T)|(k-2)}$. This conjecture concerns the Laplaci…

Convergence-equivalence conjecture. Left-convergence, partitionable convergence, and action convergence of the $(r-1)$-action of the sequence

Let $\mathcal{H}=(V,E)$ be a hypergraph, and call it $3$-uniform when every edge has size $3$. A hypergraph vertex coloring is proper when every edge contains at least two vertices…

The $B_k$-free hypergraph conjecture. One should have

Let $r\ge 2$, let $T$ be an $r$-uniform tree, and let $n$ be a positive integer. An $r$-uniform tree $T$ is $n$-good when its Ramsey number against the complete $r$-uniform hypergr…

Let $r\ge 2$, let $T_1$ and $T_2$ be $r$-uniform trees of order $m$, and let $K_n^{(r)}$ be the complete $r$-uniform hypergraph on $n$ vertices. The Ramsey equivalence conjecture f…

Peng–Zhao's conjecture. The Lagrangian of $G$ satisfies

He and Ye's free-vertex conjecture. Every $4$-regular $4$-uniform hypergraph contains a free vertex.

Let $r$ and $k$ satisfy $k=r+1>2$, and let $\mathcal{H}$ be an $r$-uniform hypergraph, meaning that every hyperedge has size $r$. A Berge path of length $k$ consists of $k$ distinc…

Let $k\geq 3$ be an integer. An $n$-vertex $k$-uniform hypergraph is said to have no $2$-regular subgraphs on $2k$ vertices if it contains no subhypergraph on exactly $2k$ vertices…

Let $f_k(n)$ denote the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no generalized $4$-cycle. For $k\geq 4$ and $n\in\mathbb{N}$, Füredi's conjecture…

Suppose that $k,\ell\in\mathbb N$ with $\ell\leq k-1$. For a $k$-uniform hypergraph $H$, let $\delta_\ell(H)$ denote its minimum $\ell$-degree, and let $m_\ell(k,n)$ be the smalles…

Let $G$ be a $k$-chromatic $r$-graph of order $n$, where $k\geq 2$ and $n>(r-1)k$. For $p\geq 1$, let $\lambda^{(p)}(G)$ be the $p$-spectral radius of $G$, and let $Q_k^r(n)$ be th…

Let $G$ be a $k$-chromatic $r$-graph of order $n$, where $k\geq 2$ and $n>(r-1)k$. For $p\geq 1$, write $\lambda^{(p)}(G)$ for the $p$-spectral radius of $G$, and let $Q_k^r(n)$ de…

Let $G$ be a $k$-uniform hypergraph on $n$ vertices. For integers $n$, $d$, $k$, and $s$ with $1\leq d\leq k-1$ and $0\leq s\leq (1-\varepsilon)n/k$, let $m_d^s(k,n)$ denote the mi…