8 problems

Let $\eta>0$ and let $r$ be a positive integer. For each $r$, an $r$-hyperedge coloring of the complete $r$-uniform hypergraph $\mathcal{K}_n^r$ assigns one of $r$ colors to every…

Let $3\leq k<n$, and let $K_n^{(k)}$ be the complete $k$-uniform hypergraph on $n$ vertices, with $\binom nk$ edges. A Hamilton Berge cycle is a Berge cycle of length $n$ using all…

Let $\mathcal{B}^r C_n$ denote the family of $r$-uniform Berge cycles of length $n$, and let $R(\mathcal{B}^r C_n;r-1)$ be the corresponding $(r-1)$-color Ramsey number. The parame…

Let a $k$-uniform covering hypergraph be a hypergraph in which every hyperedge has size $k$ and every pair of vertices is contained in a hyperedge. Let $n$ be the number of vertice…

Füredi–Kostochka–Luo conjecture. A statement similar to their exact bound for $k\geq r+3$ should hold: the extremal number and equality cases for $r$-graphs with no Berge cycle of…

Let $r\geq 3$ and let $k\geq r+3$. For an $r$-uniform hypergraph on $n$ vertices, let ${\rm EG}_r(n,k)$ denote the maximum number of edges in a hypergraph containing no Berge cycle…

Extension of the main Berge-cycle theorem. If $\mathcal H$ has no Berge cycle of length $k$ or longer, then

The long Berge-cycle bound conjecture. The same statement holds for $k=r+2$.