5 problems

Let $t$ and $n$ be positive integers, and let $\textbf{G}=\{G_1,G_2,\ldots,G_{\frac{n}{t}\binom{t}{2}}\}$ be an $n$-vertex graph system. Rainbow Hajnal--Szemerédi conjecture. If ……

Let $n,k,m$ be positive integers, and let $\textbf{H}=\{H_1,\ldots,H_m\}$ be an $n$-vertex $k$-graph system, so every $H_i$ is a $k$-uniform hypergraph on the same $n$-vertex set.…

Let $H_1,\dots,H_t$ be a collection of $n$-vertex graphs with bounded degree, let $K_n$ be the complete graph on $n$ vertices, and let $e(H_i)$ denote the number of edges of $H_i$.…

Let $K_n$ be the complete graph on $n$ vertices, and let an edge-colouring be $\mu n$-bounded when each colour appears on at most $\mu n$ edges. A rainbow Hamilton cycle is a Hamil…

Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each part, and let $K_n$ be the complete graph on $n$ vertices. An edge-colouring is proper when no two edges inc…