Minimum degree in simplicial complexes
Christian Reiher, Bjarne Schülke
TL;DR
This work resolves the extremal problem of determining the density threshold $\\alpha(d)$ that guarantees a low-degree vertex in any nonempty simplicial complex. The authors prove a sharp formula in the natural regime $d\\ge m\\ge 1$: $\\alpha(2^d-m)=\\frac{2^{d+1}-m}{d+1}$, guided by a robust local-to-global method that leverages vertex-weights, conglomerates, and a mountains framework. Central to the argument are weighted Katona–Kruskal type inequalities, refined local bounds, and a meta-theorem that aggregates local contributions into a global lower bound; these tools also yield sharp results in nearby parameter ranges (e.g., $\\alpha(11)=\\frac{53}{10}$) and illustrate the limits of the upper-bound constructions. The paper thus advances extremal set theory by providing exact thresholds in a broad natural regime, while outlining controlled extensions beyond it and presenting a complete table of known values up to $d\\le 16$ and several beyond.
Abstract
Given $d\in\mathbb{N}$, let $α(d)$ be the largest real number such that every abstract simplicial complex $\mathcal{S}$ with $0<\vert\mathcal{S}\vert\leqα(d)\vert V(\mathcal{S})\vert$ 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 $d\geq m\geq 1$, we have $α(2^d-m)=\frac{2^{d+1}-m}{d+1}$. Similar results were obtained independently by Li, Ma, and Rong.