Minimum degree in simplicial complexes

Given $d\in \mathbb{N}$, let $\alpha (d)$ be the largest real number such that every abstract simplicial complex $\mathcal{S}$ with $0<\mathrm{\mid }\mathcal{S}\mathrm{\mid }\le \alpha (d)\mathrm{\mid }V(\mathcal{S})\mathrm{\mid }$ has a vertex of degree at most $d$. We extend previous results by Frankl, Frankl and Watanabe, and Piga and Schülke by proving that for all integers $d$ and $m$ with $d\ge m\ge 1$, we have $\alpha (2^d-m)=\frac{2^{d+1}-m}{d+1}$. Similar results were obtained independently by Li, Ma, and Rong.