47 problems

For a $3$-uniform hypergraph $F$, let $\operatorname{ex}_3(n,F)$ be the maximum number of edges in an $F$-free $3$-graph on $n$ vertices, and define its Turán density by … Let…

Let $K_4^{(3)}$ be the complete $3$-uniform hypergraph on $4$ vertices, and let $K_4^-$ be its $3$-edge subgraph. Write $\pi(K_4^-)$ for the Turán density of $K_4^-$. Frankl–Füredi…

Mubayi–Rödl conjecture.

For every integer $r\ge 2$, let … Here $\pi(\mathcal{F})$ denotes the Turán density of the family $\mathcal{F}$. Chung and Graham's rationality conjecture. The set…

Erdős's conjecture.

The smallest-non-jump conjecture. As the $ABB$ pattern is the smallest one with the relevant property, $4/9$ is conjectured to be the smallest non-jump for $3$-graphs.

Mubayi's conjecture. The Turán density of $F_4$ is

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 $r$ be a positive integer. For a family $\mathcal{F}$ of $r$-uniform hypergraphs, let its Turán density be … A density $\alpha\in[0,1)$ is a jump for $r$ if there exists $c>0$…

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 a $3$-graph, and let $\pi_{\mathrm{U}}(F)$ denote its uniformly dense Turán density. In particular, $K^{(3)}_4$ is the complete $3$-graph on four vertices. Rödl's conjec…

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…

Let $r\geq 3$ and define … The minimum-positive-density conjecture. The number $\pi_r$ is the minimum positive value of the uniform Turán density of $r$-graphs. The paper proves th…

For an integer $k\geq 3$, let $\Pi_{\infty}^{(k)}$ be the set of Turán densities of families of $k$-uniform hypergraphs: … A subset of the real numbers is well-ordered if every non…

Let $K_4^{(3)}$ be the complete $3$-uniform hypergraph on four vertices. Its Turán density $[?]$ is the limit of the maximum edge density of $K_4^{(3)}$-free $3$-graphs. Turán dens…

Daisy density conjecture for three-element petals.

Bollobás–Leader–Malvenuto–Bukh conjecture.

The minimum-density conjecture. If $F$ has no vanishing ordering of $V(F)$ and, for every pair $\{i,j\}\in [k]^2$ with $i<j$, it can be partitioned into spanning subhypergraphs…

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 $r>k\ge 2$ and $e\ge 3$ be fixed integers. For an $n$-vertex $r$-uniform hypergraph, let $f_r(n,v,e)$ denote the maximum number of edges such that the union of any $e$ distinct…

Lagrangian-density conjecture. The Lagrangian density of $K_4^3\cup e$ is attained by $S_2(n)$ as $n\to\infty$. The conjecture is proposed as an approach to the unresolved Turán-de…

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…

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…