Minimum Degree Thresholds for Simplicial Complexes in Extremal Set Theory
An extremal-combinatorics result giving the exact edge-to-vertex ratio below which an abstract simplicial complex must contain a low-degree vertex. Writing alpha(d) for the largest constant such that every complex with at most alpha(d) times as many edges as vertices has a vertex of degree at most d, the work proves alpha(2d - m) = (2^{d+1} - m)/(d+1) for all integers d >= m >= 1, extending prior partial results, and separately proves alpha(11) = 53/10, settling a conjecture of Frankl and Watanabe.
Minimum Degree in Simplicial Complexes
A pure-combinatorics paper in extremal set theory about abstract simplicial complexes (finite set systems closed under taking subsets). For a degree bound d, alpha(d) is defined as the largest consta…