8 problems

Trotter's conjecture. For every fixed $Delta$ and $r$, the induced Ramsey number $r_{\operatorname{ind}}(H;r)$ is polynomial in $n$ for every $n$-vertex graph $H$ of maximum degree…

Let $G$ be a graph with $n$ vertices, and let $\operatorname{IR}(G,G)$ denote the minimum number of vertices in a graph that strongly arrows $(G,G)$ under every red-blue edge-colou…

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 $G$ be a graph with $n$ vertices, and let $r_{\mathrm{ind}}(G,G)$ denote the induced Ramsey number for two copies of $G$. Erdős's conjecture. There is a positive constant $c$ s…

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…

Let $H$ be an $n$-vertex graph, and let $\hat{r}^2_{\operatorname{ind}}(H)$ denote the smallest number of edges in a graph whose every $2$-coloring contains a monochromatic copy of…