2 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 $\eta(G)\leq\omega(G)^d$.