4 problems

Let $G$ be a fixed graph, and let $H$ be a graph on $n$ vertices. The quantity $r_{\mathrm{ind}}(G,H)$ is the minimum order of a graph whose every red-blue edge-coloring contains e…

Let $H$ be a graph with $k$ vertices, and let $R^{\mathrm{ind}}(H)$ denote the minimum number of vertices of a graph $G$ such that every red-blue edge-colouring of $G$ contains an…

Let $k$ be a positive integer. Exponential gadget conjecture. There is a graph $G$ with $e^{O(k)}$ edges such that every $k$-coloring of its edges contains a monochromatic odd cycl…

Let $C_n$ be the cycle on $n$ vertices, and let $\hat{r}^k_{\operatorname{ind}}(C_n)$ denote the smallest number of edges in a graph whose every $k$-coloring contains a monochromat…