Sharp Bounds on the Independence Number of Simplicial Spheres

We study the maximum size of an independent set in the graph of a simplicial sphere. Let $\beta(d,n)$ denote this maximum over all simplicial $(d-1)$-spheres on $n$ vertices, and let $\alpha(d,n)$ denote the maximum restricted to flag $(d-1)$-spheres. For every fixed $d\geq4$, we prove $\beta(d,n)=n-\Theta(n^{1/\lfloor d/2\rfloor})$. For flag spheres, we show $\alpha(d,n)\geq n-4\sqrt n+O(1)$ for all $d\geq4$ and determine the correct asymptotic order $\alpha(d,n)=n-\Theta(\sqrt n)$ for dimensions $d=4,5$. We also investigate the independence sets of Bier spheres and show that, in contrast to our other results, for this very large family of spheres, the independence number cannot be larger than $\left\lfloor\frac{n}{2}\right\rfloor.$