7 problems

Let $G$ be a loopless simple graph, let $\chi(G)$ denote its chromatic number, and let $K_t$ be the complete graph on $t$ vertices. A strong immersion of $K_t$ in $G$ is an immersi…

All graphs considered are finite and simple. For a graph $G$, write $\rho(G)$ for its chromatic number and $\tau(G)$ for its clique number. A graph is minimally nonperfectly divisi…

A graph $H$ is weakly $K_k$-Turán-good when, for all sufficiently large $n$, some complete $(k-1)$-partite graph attains ${\mathrm{ex}}(n,H,K_k)$. An edge of a graph $F$ is color-c…

Let $G$ be a graph, let $\chi(G)$ denote its chromatic number, and let $K_t$ be the complete graph on $t$ vertices. An $H$-immersion is an injective mapping of $V(H)$ to $V(G)$ tog…

For a graph $F$, let $cchi(F)$ denote its chromatic number. Define the $F$-decomposition threshold $cdelta_F$ as the limsup, over the number $n$ of vertices, of the minimum relativ…

Conjecture on consecutive holes. Every graph with huge chromatic number and bounded clique number contains $k$ holes with consecutive lengths.

Let $G$ be a loopless $d$-chromatic graph, and let $K_d$ denote the complete graph on $d$ vertices. A graph $H$ is a minor of $G$ if a graph isomorphic to $H$ can be obtained from…