Polynomial-expansion path-degeneracy lower-bound conjecture

Let $b>0$ be a real number. A graph class has polynomial expansion of exponent $b$ if its expansion is $O(r^b)$. A graph is $p$-path degenerate if it admits the path-degeneracy property used in the paper. The girth of a graph $G$ is denoted by $\operatorname{girth}(G)$.

Polynomial-expansion lower-bound conjecture. For every real $b>0$, there exists a graph class with expansion $O(r^b)$ such that for infinitely many $p\in\mathbb N$ the class contains a graph $G$ satisfying

$$\operatorname{girth}(G)\sim 4bp\log_2 p$$

that is not $p$-path degenerate.

The preceding theorem gives a lower bound with leading term $(8/3)bp\log_2p$, while the proved upper bound has leading term $4bp\log_2p$. The conjecture asserts that the upper bound is asymptotically sharp up to the second-order term; the stated cubic-cage conjecture would imply it.