7 problems

Cambie's conjecture. Then

Füredi–Kahn–Seymour 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 $f_r(n,p,q)$ 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…

Let $k$ and $r$ be positive integers. A $k$-uniform hypergraph is a hypergraph whose edges are $k$-element subsets of its vertex set, and a tight cycle is a cyclically ordered sequ…

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…