3 problems

Matching-free polynomial eta-boundedness conjecture. For every integer $t\geq 3$, $M_t$-free graphs are polynomially $\eta$-bounded.

Polynomial $P_5$-free eta-boundedness conjecture. There exists $d\in\mathbb{N}$ such that every $P_5$-free graph $G$ satisfies $\eta(G)\leq\omega(G)^d$.

Gyárfás–Sumner analogue for $\eta$-boundedness. For every forest $H$, there exists a function $f:\mathbb{N}\to\mathbb{N}$ such that every $H$-free graph $G$ satisfies…