6 problems

Path 3-colorability conjecture. Every path is matroidally $3$-colorable.

Maximum independent edge set conjecture.

Let $P_n$ be the path on $n$ vertices, and let $\overset{2}{\uplus}~P_n$ denote its double blow-up, with corresponding vertices $(0,u)$ and $(1,u)$ for each vertex $u$ of $P_n$. Le…

Sum-diameter conjecture for paths. For $n\geq 3$, we have

Spum conjecture for paths. For $n\geq 8$, we have

Let $t$ be a positive integer, let $P_{2t}$ denote the path of length $2t$, and let $\operatorname{ex}^{-1}(k,H)$ denote the inverse Turán number of $H$, namely the maximum number…