2 problems

Let $k\geq 1$ and $m\geq 2$. For a graph $H$, let $GR_k(H)$ denote the least order of a complete graph such that every exact $k$-edge-coloring contains a monochromatic copy of $H$,…

Let $n\ge3$ and $k\ge1$. For $i\in\{0,1,\ldots,n-2\}$, let $G_i=P_{2i+3}$, and let $G_{n-1}\in\{C_{2n},P_{2n+1}\}$. Choose $i_j\in\{0,1,\ldots,n-1\}$ for $j\in\{1,\ldots,k\}$ with…