Minimum degree in simplicial complexes

Abstract

Given $d\in \mathbb{N}$, let $\hat{I}\pm (d)$ be the largest real number such that every abstract simplicial complex $\mathcal{S}$ with $0<|\mathcal{S}|\le \hat{I}\pm (d)|V(\mathcal{S})|$ 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 $\hat{I}\pm (2^d-m)=\frac{2^{d+1}-m}{d+1}$. Similar results were obtained independently by Li, Ma, and Rong.