2023/10/03 by David Conlon, Conlon, David, Simón Piga +3 · 1 citation
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2310.01822
A simplicial complex H consists of a pair of sets (V,E) where V is a set of vertices and E⊆\mathscrP(V) is a collection of subsets of V closed under taking subsets. Given a simplicial complex F and n∈ \mathbb N, the extremal number ex(n,F) is the maximum number of edges that a simplicial complex on n vertices can have without containing a copy of F. We initiate the systematic study of extremal numbers in this context by asymptotically determining the extremal numbers of several natural simplicial complexes. In particular, we asymptotically determine the extremal number of a simplicial complex for which the extremal example has more than one incomplete layer.