3 problems

For a family $\mathcal{F}$ of graphs, a graph $G$ is $\mathcal{F}$-free if no induced subgraph of $G$ is isomorphic to a graph in $\mathcal{F}$. Let $P_r$ be the path with $r$ vert…

Let $k\ge 3$, let $\varepsilon>0$, and let $G$ be an $n$-vertex incomparability graph. Here $\varrho_2(G)$ denotes the edge density of $G$, and $K_k[t]$ denotes the complete $k$-pa…

Let $H$ be a graph, and for each integer $s$ let $p(s,H)$ be such that every graph without a copy of $K_{s,s}$ and with average degree at least $p(s,H)$ contains an induced subdivi…