74 problems

Erdős–Faber–Lovász conjecture. The vertices of $\mathcal{H}$ can be colored properly with $n$ colors, so that no edge contains two vertices of the same color.

Let $q\geqslant 2$ and let $\mathrm{KG}^q(n,q)$ be the $q$-uniform Kneser hypergraph on the $q$-subsets of $[n]$, with edges consisting of $q$ pairwise disjoint vertices. Let…

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…

Zhu's conjecture. For given hypergraphs $\mathcal{H}_1$ and $\mathcal{H}_2$,

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

Given an integer $k\ge 2$, let $\chi'_k(G)$ denote the minimum number of edge-disjoint subgraphs into which the edge set of a graph $G$ can be partitioned so that every vertex has…

Erdős's conjecture. For $r>r_0(n)$,

Let $m(n,r)$ be the minimal number of edges in an $n$-uniform hypergraph with chromatic number more than $r$. Erdős conjectured that, for each fixed $n$, there is a threshold…

Erdős–Lovász conjecture. Neither the lower bound $m_2(r)=\Omega(\sqrt{r/\log r}\,2^r)$ nor the upper bound $m_2(r)=O(r^2 2^r)$ is sharp, and the correct order of magnitude is

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

Let $H_k(n,m)$ be a uniformly random $k$-uniform hypergraph on $n$ vertices with $m$ hyperedges, and let $c$ denote the hyperedge-to-vertex ratio. A hypergraph is 2-colorable if it…

Kahn's conjecture. As the relevant degree parameters tend to infinity, the chromatic index of a $k$-uniform hypergraph is asymptotically equivalent to its fractional chromatic inde…

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…

Let $r$ be the uniformity, let $H$ be a sufficiently dense $r$-uniform hypergraph, and let $\rho(H)$ denote the density parameter used in the paper. Sharp threshold conjecture. The…

Let $r$ be the uniformity, let $H$ be a sufficiently dense $r$-uniform hypergraph, and let $\rho(H)$ denote the parameter used to describe its density. Sharp threshold conjecture.…

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…

Keevash's conjecture. Every $(t,r,n)$-Steiner system has chromatic index

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…