Positive co-degree density of hypergraphs

2022/07/12 by Halfpap, Anastasia, Lemons, Nathan, Palmer, Cory · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.05639

Abstract

The minimum positive co-degree of a non-empty r-graph H, denoted δ_r-1⁺( H), is the maximum k such that if S is an (r-1)-set contained in a hyperedge of H, then S is contained in at least k distinct hyperedges of H. Given an r-graph F, we introduce the positive co-degree Turán number co⁺ex(n, F) as the maximum positive co-degree δ_r-1⁺(H) over all n-vertex r-graphs H that do not contain F as a subhypergraph. In this paper we concentrate on the behavior of co⁺ex(n, F) for 3-graphs F. In particular, we determine asymptotics and bounds for several well-known concrete 3-graphs F (e.g. K₄^- and the Fano plane). We also show that, for r-graphs, the limit γ⁺(F) := lim_{n → ∞} \fracco⁺ex(n, F)n exists, and ``jumps'' from 0 to 1/r, i.e., it never takes on values in the interval (0,1/r). Moreover, we characterize which r-graphs F have γ⁺(F)=0. Our motivation comes primarily from the study of (ordinary) co-degree Turán numbers where a number of results have been proved that inspire our results.

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