3 problems

Let $K_n^k$ be the complete $k$-uniform hypergraph on $n$ vertices, with its edges colored using two colors. A loose path is a hypergraph path in which consecutive edges intersect…

Let $P_3^{(3)}$ be the 3-uniform loose path of length $3$, and let $R(P_3^{(3)},r)$ be the minimum $n$ such that every $r$-edge-coloring of the complete 3-uniform hypergraph…

Let $P_\ell^{(k)}$ be the $k$-uniform loose path of length $\ell$, and let $R(P_\ell^{(k)},2)$ denote the minimum $n$ such that every red-blue edge-coloring of $K_n^{(k)}$ contains…