Minimum degree in simplicial complexes

TL;DR

This paper determines the maximum proportion of simplicial complexes that guarantees the existence of a vertex with degree at most a given value, extending previous bounds with a precise formula.

Contribution

It provides an exact formula for the threshold proportion in simplicial complexes ensuring low-degree vertices, generalizing earlier partial results.

Findings

Established the value of α(2^d - m) as (2^{d+1} - m)/(d+1)

Extended previous bounds on vertex degrees in simplicial complexes

Confirmed similar results independently obtained by other researchers

Abstract

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