195 problems

For every fixed integer $k\ge 2$, there exists a constant $C_k>0$ such that every $3$-uniform hypergraph $H$ with $m$ edges has a partition $V(H)=V_1\sqcup\cdots\sqcup V_k$ satisfy…

For every positive integer $n$ and every collection of independent nonnegative random variables $X_1,\ldots,X_n$ satisfying $\mathbb{E}[X_i]=1$ for all $i$, one has…

For every integer $k\ge 2$, there exists a convex function $f_k:[0,\infty)\to[0,1]$ satisfying … as $x\to\infty$, such that every uncrowded $k$-uniform hypergraph $H$ admits a loca…

For every fixed pair of integers $r,k\ge 3$, let $C_k^{(r)}$ denote the $r$-uniform linear cycle with $k$ hyperedges, and let $\operatorname{ex}_r(n,C_k^{(r)})$ be the maximum numb…

For every fixed $\varepsilon>0$, determine whether the random graph $G(n,p)$ has, with high probability, a fractional triangle decomposition precisely at the conjectured threshold…

For every fixed arity $k$, every $k$-ary constraint-satisfaction problem having a Mal'tsev extension has linear non-redundancy; equivalently, its non-redundancy is $O(n)$ in the re…

For integers $r,k\ge 2$ and $s\ge r$, let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $r$-uniform hypergraph $H$ on $n$ vertices such that no set of $k$ distinct edges ha…

For the tetrahedron $K_4^{(3)}$, determine whether its Turán density satisfies $\pi(K_4^{(3)})=\frac{5}{9}$. Equivalently, if $\operatorname{ex}(n,K_4^{(3)})$ denotes the maximum n…

For every integer $r\ge 2$ and every integer $d\ge 1$, if $H$ is a finite simple $r$-uniform $d$-regular hypergraph, then the number of weak independent sets of $H$ satisfies…

Determine whether the uniform Turán density of the tetrahedron satisfies $\pi_u\!\left(K_4^{(3)}\right)=\frac{1}{2}$. Here $K_4^{(3)}$ is the complete $3$-uniform hypergraph on $4$…

Let $k\ge 3$ be fixed and let $H$ be a $k$-uniform hypergraph on $n$ vertices. For each $(k-2)$-set $S\subseteq V(H)$, let $L_H(S)$ be the link graph of $S$, whose vertices are…

For every $3$-uniform hypergraph $F$, $\pi_{\mathrm{co}}(F)=0$ if and only if $F$ is layered and $\pi(F)=0$, where $\pi(F)$ is the uniform Turán density and…

For every $d>0$ and every $\alpha>0$, do there exist $\rho>0$ and $n_0$ such that every $(\rho,d)$-dense $3$-uniform hypergraph $H$ on $n\ge n_0$ vertices with minimum codegree…

Feige's hypergraph Moore bound conjecture. For every $k\ge 3$, there exist constants $c_k,C_k>0$ such that, whenever

Turán's conjecture. For any integer $k\geqslant 4$, we have

Frankl–Füredi conjecture. If $G$ is an $r$-graph with $m$ edges, then

Let $\mathcal{P}^k_n$ and $\mathcal{C}^k_n$ denote the $k$-uniform loose path and loose cycle with $n$ edges, respectively, and let $R(H_1,H_2)$ be the two-colour Ramsey number for…

Let $H$ be an $r$-partite hypergraph containing at least one edge, and let $\nu(H)$ denote its matching number, the maximum number of pairwise disjoint edges. For a set $S$ of vert…

Large-degree conjecture. There exists $d_r>0$ such that if $H$ has an edge of multiplicity at least $\nu+1$, then

Han–Zhao's conjecture. If

Strong maximality and minimality conjecture. Every such hypergraph has a strongly maximal matching, a strongly minimal vertex cover, and a strongly minimal edge cover.

Mubayi's conjecture. If $\mathcal F$ is a stable non-$r$-partite $r$-graph, then, for every positive integer $q$ and all sufficiently large $n$, every $n$-vertex $r$-graph with…

Hypergraph Nash–Williams–Tutte conjecture. For positive integers $t$ and $k$, every $k$-weakly-partition-connected hypergraph $H$ on $t$ vertices has a $k$-distinguishable tree ass…

Let $k\geq 4$ and $0<\delta\leq 1$. A linear $3$-graph is a $3$-uniform hypergraph in which any two edges share at most one vertex; for a linear $3$-graph $\mathcal{H}$ with $n$ ve…

Let $\mathcal{C}_5^3$ be the $3$-uniform tight cycle of length $5$, with vertex set $\{1,\dots,5\}$ and edges of the form $\{x,x+1,x+2\pmod 5\}$. Let $\pi(\mathcal{C}_5^3)$ denote…