25 problems

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…

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…

For each integer $k$ with $4\le k\le 10$, let $S_k$ denote the star $3$-graph considered in the paper, and let $\pi_{\mathrm{U}}(S_k)$ be its uniformly dense Turán density. Conject…

Let $F$ be an $r$-graph. An ordering $\preceq$ on $V(F)$ and a function … assign to each pair of vertices the role it plays in an edge containing that pair. The zero-density charac…

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

For integers $r\ge 2$ and sufficiently large integers $p$, let $t_r(2p+1,p+1)$ denote the local Turán density of the complete $r$-uniform hypergraph on $p+1$ vertices in a hypergra…

Mubayi–Pikhurko–Sudakov conjecture. If $\ell\geq 5$ is not divisible by three, then

Let $\mathcal{C}_5^3$ be the $3$-uniform tight cycle on $5$ vertices, and let $\mathcal{C}_5^{-}$ be the $3$-uniform hypergraph obtained from $\mathcal{C}_5^3$ by removing one edge…

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…

For an integer $r\ge 2$, an $r$-graph is an $r$-uniform hypergraph. A real number $\alpha\in[0,1]$ is a jump for $r$ if there exists a constant $c>0$ such that, for every…

For an integer $r\geq 2$, let $\Pi_{1}^{(r)}$ denote the set of Turán densities of single $r$-graphs. The normalized Turán-density conjecture. If … is in $\Pi_{1}^{(r)}$ for…

Let $X$ be a finite set, and let $\mathcal{H}\subseteq\binom{X}{r}$ be an $r$-uniform hypergraph. For integers $q\geq p\geq r\geq2$, say that $\mathcal{H}$ has the $(q,p)$-property…

Let $t_r(q,p)=\lim_{n\to\infty}T_r(n,q,p)/\binom{n}{r}$, where $T_r(n,q,p)$ is the minimum number of edges in an $r$-uniform hypergraph on $n$ vertices having the property that eve…

Let $T'$ and $T$ be $r$-ary generalized directed hypergraph theories for $r \geq 3$. Write $J_T$ for the set of jumps of $T$ and $m_T$ for its multiplicity parameter. Jump-separati…

For $s\geq2$, let $K_s^{(2)}$ be the complete $2$-graph on $s$ vertices, let $S(K_s^{(2)})$ be its suspension, and let $4\pi(S(K_s^{(2)}))4$ denote its Turán density. Asymptotic su…

Iterated-suspension limit conjecture. For all hypergraphs $H$,

Iterated-suspension conjecture. For any $k\geq 2$,

The $4.2$-density conjecture.

The $K_4^-$-density conjecture.

The $4.2$-density conjecture.

Frankl–Füredi–Razborov conjecture.

The $K_4^-$-density conjecture.

Keevash–Mubayi conjecture. $G_t(n)$ is the unique, up to isomorphism, $3$-graph with $\operatorname{ex}(n,K_t)$ edges and $\operatorname{ex}_{K_t^-}(n,K_t)$ induced copies of…

Let $\pi(\operatorname{odd\ cycle\ in\ link\ graph},F_{3,2})$ denote the maximum asymptotic edge density of a 3-graph containing no copy of $F_{3,2}$ and having no odd cycle in any…

A 3-graph is a 3-uniform hypergraph, and the link graph of a vertex is the graph whose vertices are the remaining vertices, with an edge for each 3-edge containing the given vertex…