35 problems

Asymptotic lower-bound conjecture. For $k\geq r+1$,

Alon–Kim conjecture. For every $k\ge t\ge 1$ and $\varepsilon>0$, there exists $D_0$ such that, for every $D\ge D_0$, every $k$-uniform, $t$-simple hypergraph with maximum degree a…

Let $r\geq 3$ be a fixed integer. For a path $P_k$ with $k$ edges, let $\mathcal{B}(P_k)$ denote its Berge hypergraph, and let $\mathrm{schex}_r(n,\mathcal{B}(P_k))$ be the strong…

Dhawan's conjecture. For every $k\geqslant 2$, there is a constant $c=c(k)>0$ such that, for all sufficiently large $\Delta$,

Let $H$ be a hypergraph. Write $\chi_p(H)$ for its panchromatic number, $\chi^2_p(H)$ for its bipanchromatic number, and let $\alpha_{\chi_p(H)}(H)$ be the minimum number of unique…

Let $\mathcal F$ be a hereditary family of hypergraphs, and let $m_k(\mathcal F)$ be the least threshold guaranteeing a polychromatic $k$-coloring for every $m$-heavy member of…

Let $\mathcal F$ be a hereditary family of hypergraphs. For $k\ge1$, let $m_k(\mathcal F)$ be the smallest integer $m$ such that every $m$-heavy hypergraph in $\mathcal F$ has a po…

Asymptotic chromatic-number conjecture. For any $r\geq 2$, there exists an integer $n_0(r)$ such that, for all $n\geq n_0(r)$,

Berge–Füredi conjecture. A linear (loopless) hypergraph $(V,E)$ satisfies

Let $[n]=\{1,\ldots,n\}$, and let $\mathrm{KG}^r(n,k)_{s\textup{-stab}}$ be the induced $r$-uniform Kneser hypergraph whose vertices are the $s$-stable $k$-subsets of $[n]$, where…

Keszegh–Pálvölgyi conjecture. If every hyperedge has more tail-vertices than head-vertices, and for every $e_{1},e_{2}\in E(H)$ with $|e_{1}\cap e_{2}|=1$ the common vertex is a he…

Let $k\geqslant 2$, let $\Delta$ be sufficiently large, and let $H$ be a $k$-partite $k$-graph of maximum degree at most $\Delta$. Write $\chi_\ell(H)$ for the list chromatic numbe…

Colorability-extension conjecture. For any $r,m,n\in \mathbb{N}$ with $r\ge 3$ and $n\ge (r-1)^2+1$, if all bi-hypergraphs in $Bi\mathcal{H}yp(n,r,m)$ are colorable, then all bi-hy…

Minimum-size conjecture. For any $r\in \mathbb{N}$ with $r\ge 4$,

Conjecture on $\ell(7,3)$.

Existence conjecture. For any $n,r\in \mathbb{N}$ with $r\ge 3$ and $n\ge (r-1)^2+1$, the set $\mathcal{MUC}(n,r)$ is not empty.

Keszegh–Pálvölgyi linear bound conjecture. For every $k$ and every hereditary family $\mathcal F$,

Polychromatic finiteness conjecture. If $m_2(\mathcal F)<\infty$, then $m_k(\mathcal F)<\infty$ for every $k$ for any hereditary family $\mathcal F$.

Let $k,\Delta\geq 2$. A harmonious coloring of a $k$-uniform hypergraph $H$ is a vertex coloring that is rainbow on every edge and assigns distinct color sets to distinct edges; le…

For an integer $t\geq 2$, let $h(t)$ be the largest chromatic number of a hypergraph with no $K_t$-minor; the paper establishes that this quantity exists. A hypergraph is $K_t$-min…

Let $r\geq 2$ and let $\mathcal{F}$ be a set system. Write $\mathcal{F}_{\widetilde{r-\operatorname{stab}}}$ for the family obtained by retaining the source's weakened notion of…

Matsumoto–Ohno conjecture. If $H$ is $3$-colorable, then it admits a complete $t$-coloring for every integer $t$ satisfying

Berge–Füredi–Meyniel conjecture. For any linear hypergraph $H$, its chromatic index is at most $\Delta^*(H)+1$.

For integers $2\leq k<l$, let $\mathbf{LO}_k$ and $\mathbf{LO}_l$ denote the corresponding symmetric relational templates, and let $\mathrm{PCSP}(\mathbf{LO}_k,\mathbf{LO}_l)$ be t…

Let $r\geq 2$. An injective $q$-coloring of an $r$-uniform hypergraph is a coloring of its vertices with $q$ colors such that any two edges sharing $r-1$ vertices have distinct col…