184 problems

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…

Jump threshold conjecture. All numbers in $[0,4/9)$ are jumps and all numbers in $[4/9,1)$ are non-jumps.

Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no collection of $k$ edges spanning at most $s$ vertices. For positive intege…

Let ${\cal H}=(V,{cal E})$ be a hypergraph with $|V|=n$, no isolated vertices, and every edge $h\in{cal E}$ of size $r$. Let $\sigma_v({\cal H})$ and $\alpha({\cal H})$ be the stated…

Simultaneous multipartite subhypergraph conjecture. There exists a partition of $V$ into $r$ classes $V_1,\dots,V_r$ such that, for every $i=1,\dots,\ell$, at least

The covering-number converse conjecture. If

Let $H$ be a red/blue coloured $3$-graph on $n$ vertices, and let $\delta_1(H)$ denote its minimum vertex degree. A loose Hamilton cycle is a cyclic ordering of the vertices in whi…

Erde–Kang–Lehner–Mohar–Schmid conjecture.

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…

Frankl–Gryaznov–Talebanfard's conjecture. If $\mathcal{H}$ contains no clique of size $ell+1$, then

Let $\mathcal{H}$ be an $r$-graph, and let $d_{\mathcal{H}}(e)$ denote the codegree of an $(r-1)$-set $e$. Define … For an $r$-graph $\mathcal{H}$, let … Given a balanced partition…

Critical-excess conjecture. If

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 $r\ge 1$ and $s\ge 1$ be integers, and let $H$ be an $r$-uniform hypergraph with $s(H)=s$. Asymptotic equivalence conjecture. Then … The quantity…

Let $r\ge 2$, $s\ge 2$, and $k\ge r+1$ be integers. Let … Let $H$ be an $r$-uniform hypergraph with $s(H)=s$ and $\delta_{s-1}(H)=\delta$. The gamma lower-bound conjecture. Then ……

Let $H$ be a $3$-graph on $n$ vertices, and let $\delta_2(H)$ denote its minimum codegree, the minimum number of vertices completing a pair to an edge of $H$. The square of a tight…

Han–Zhao's conjecture. If

Let $P_4^r$ be the $r$-uniform linear path with four edges, and let $ex_r^{\mathrm{lin}}(n,P_4^r)$ denote the maximum number of edges in an $n$-vertex linear $r$-uniform hypergraph…

Let $k$-graphs be $k$-uniform hypergraphs, let $\beta(F)$ denote the codegree Turán density of a $k$-graph $F$, and let $\beta(\rho)$ denote the corresponding density for a family…

Acyclic $(3,4)$-tournament spanning-path conjecture. If $G$ is a $(3,4)$-tournament with no closed walk, then $G$ has a spanning path.

$(3,4)$-tournament spanning-path conjecture. Every $(3,4)$-tournament has a spanning path. That is, $f(n,3,4)=n$.

Let $Y$ be a Young diagram and let $H(Y)$ be the associated 3-partite 3-uniform hypergraph. Write $|Y|$ for the number of cells of $Y$, let $\tau^{(2)}(H(Y))$ denote its second cov…

Let $H$ be an $r$-partite $r$-graph, let $A$ be one of its $r$ partition classes, and let $\mathbf{RG}_{\mathrm{Mat}}(H,A;k)$ denote the matching reconfiguration graph on matchings…

Let $S_4$ be the four-vertex star $3$-graph, and let $\pi_{\mathrm{V}}(S_4)$ denote the corresponding Turán-density parameter. Conjecture for the four-vertex star. … This is the se…