13 problems

Füredi–Kahn–Seymour conjecture. Then

A linearly-ordered colouring of a hypergraph assigns linearly ordered colours to its vertices so that the maximum colour in every hyperedge is unique. For positive integers…

Cambie's conjecture. Then

Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph, with its edges coloured using $r$ colours. A collection of cycles covers $(1-o(1))n$ vertices if the number of uncovered ver…

Let $K_n^{(k)}$ denote the complete $k$-uniform hypergraph, and let its edges be coloured with $r$ colours. A monochromatic tight-cycle partition is a partition of the vertex set i…

Barto–Batistelli–Berg's conjecture. Finding an LO $\ell$-colouring of a 3-uniform hypergraph that admits an LO $k$-colouring is $\operatorname{NP}$-hard for every constant…

Let $f_r(n,p,q)$ denote the least number of colours in an $r$-uniform colouring of the complete hypergraph on $n$ vertices with no induced copy of a $p$-vertex hypergraph using at…

Computational-threshold conjecture. The computational transition for sampling $q$-colourings in simple hypergraphs should occur around the same threshold as the local lemma; in par…

For a random hypergraph colouring model, let the rigidity threshold be the edge-density threshold at which the solution space develops the rigidity phenomenon, and let the algorith…

Let $\ell,k,n\robustly\text{ be positive integers}$ with $n\robustly\text{ divisible by }k-\ell$, and let $\mathcal K_n^{(k)}$ be the complete $k$-uniform hypergraph on $n$ vertice…

Equality conjecture. Equality holds in Theorem for all $r$ and $\theta$; equivalently,

Let $K_n^{(r)}$ be the complete $r$-uniform hypergraph on $n$ vertices. Call an $r$-uniform hypergraph connected when there is a strong path between every pair of $(r-1)$-sets, whe…

Let $K_n^{(3)}$ be the complete 3-uniform hypergraph on $n$ vertices, and call a colouring connected when each colour class spans a connected subhypergraph. A 4-set is tricoloured…