4 problems

For a finite simple graph $G$, let $N(G,H)$ be the number of unlabeled, not necessarily induced copies of $H$ in $G$. Define … Let $P_k$ denote the path on $k$ vertices, so…

Odd-length extremal conjecture. For any odd integer $\ell\geq 3$ and sufficiently large $n$, it holds that

Gerbner–Palmer's conjecture. Every path is $K_{k+1}$-Turán-good for every $k$.

Let $G$ be a connected graph and let $u,v\tnobreak$ be vertices with $d(u),d(v)\geq 2$. Suppose that $u$ and $v$ are joined by a path … where $d(w_i)=2$ for $1\leq i\leq s-1$.…