4 problems

Kalai's conjecture. If

Let $q>r\geq 3$ be integers. An $r$-graph is an $r$-uniform hypergraph, and it is $K_q^r$-divisible when it satisfies the divisibility conditions necessary for a decomposition into…

Localized hypergraph clique-weight conjecture. One has

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…