Large Independent Sets in Flag Spheres

For every $d\ge 4$, we construct a family of $(d-1)$-dimensional flag simplicial spheres ${\mathcal{K}}_n$ whose graphs contain independent sets of size asymptotically equal to the number of vertices. More precisely, we prove that for sufficiently large $n$, $$\alpha (G({\mathcal{K}}_n))\ge f_0({\mathcal{K}}_n)-\frac{C{f}_0({\mathcal{K}}_n)}{{(\log f_0({\mathcal{K}}_n))}^{\lfloor d\mathrm{/}2\rfloor -1}},$$ where $C=C(d)>0$. This disproves a recent conjecture of Chudnovsky and Nevo.