Positive co-degree density of hypergraphs
Anastasia Halfpap, Nathan Lemons, Cory Palmer
TL;DR
The paper introduces and analyzes the positive co-degree Turán framework for r-graphs, defining δ_{r-1}^+(H) and the quantities co^+ex(n,F) and γ^+(F). It proves the existence of the limit γ^+(F) and reveals a sharp dichotomy: for r-graphs, γ^+(F) either equals 0 (when F is r-partite) or is at least 1/r, with a jump from 0 to 1/r. For 3-graphs, the authors determine tight bounds or exact values of co^+ex(n,F) for several small forbidden graphs (K_4^−, F_5, F_{3,2}, F_{3,3}, C_5, C_5^−, J_k) and provide constructions and supersaturation results guiding these bounds. They also establish general principles such as blow-up invariance and a removal-based approach that yield the existence of γ^+(F) and underpin the 0–1/r jump phenomenon. The results advance understanding of degree-version Turán problems and connect to classical co-degree extremal theory, with broader implications for hypergraph extremal problems and density jumps in higher uniformities.
Abstract
The \emph{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 \emph{positive co-degree Turán number} $\mathrm{co^+ex}(n, {F})$ 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 $\mathrm{co^+ex}(n, {F})$ 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 $r$-graphs, the limit \[ γ^+(F) := \lim_{n \rightarrow \infty} \frac{\mathrm{co^+ex}(n, {F})}{n} \] 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.