Independence-number rationality conjecture for FI-graphs

Let $G_{\bullet}$ be an $\operatorname{FI}$-graph, and let $\alpha(G_n)$ denote its independence number. Define the generating function

$$\sum_{n \geq 0} \alpha(G_n)t^n.$$

Independence-number rationality conjecture. The generating function is rational. In particular, the function $n \mapsto \alpha(G_n)$ agrees with a quasi-polynomial for all $n \gg 0$.

This conjecture asks whether the extremal invariant given by independence numbers exhibits the same eventual regularity as counting invariants of $\operatorname{FI}$-graphs. The paper provides experimental evidence, including a classification of $\operatorname{FI}$-graphs with quadratically growing vertex sets, but no resolution is stated.