Positive co-degree density of hypergraphs

• Published in: arXiv.org 2022-07
• Main Authors: Halfpap, Anastasia, Lemons, Nathan, Palmer, Cory
• Format: Article
• Language: English
• Subjects: Graphs
• ISSN: 2331-8422

The minimum positive co-degree of a non-empty \(r\)-graph \({H}\), denoted \(\delta_{r-1}^+( {H})\), 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 a family \({F}\) of \(r\)-graphs, we introduce the {\it positive co-degree Turán number} \(\mathrm{co^+ex}(n, {F})\) as the maximum positive co-degree \(\delta_{r-1}^+(H)\) over all \(n\)-vertex \(r\)-graphs \(H\) that do not contain \(F\) as a subhypergraph. In this paper we concentrate on the behavior of \(\mathrm{co^+ex}(n, {F})\) for \(3\)-graphs \(F\). In particular, we determine asymptotics and bounds for several well-known concrete \(3\)-graphs \(F\) (e.g.\ \(K_4^-\) and the Fano plane). We also show that, for \(3\)-graphs, the limit \[ \gamma^+(F) := \limsup_{n \rightarrow \infty} \frac{\mathrm{co^+ex}(n, {F})}{n} \] ``jumps'' from \(0\) to \(1/3\), i.e., it never takes on values in the interval \((0,1/3)\), and we characterize which \(3\)-graphs \(F\) have \(\gamma^+(F)=0\). 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.