Positive Co-Degree Density of Hypergraphs

The minimum positive co-degree of a nonempty (Formula presented.) -graph (Formula presented.), denoted (Formula presented.), is the maximum (Formula presented.) such that if (Formula presented.) is an (Formula presented.) -set contained in a hyperedge of (Formula presented.), then (Formula presented.) is contained in at least (Formula presented.) distinct hyperedges of (Formula presented.). Given an (Formula presented.) -graph (Formula presented.), we introduce the positive co-degree Turán number (Formula presented.) as the maximum positive co-degree (Formula presented.) over all (Formula presented.) -vertex (Formula presented.) -graphs (Formula presented.) that do not contain (Formula presented.) as a subhypergraph. In this paper, we concentrate on the behavior of (Formula presented.) for 3-graphs (Formula presented.). In particular, we determine asymptotics and bounds for several well-known concrete 3-graphs (Formula presented.) (e.g. (Formula presented.) and the Fano plane). We also show that, for (Formula presented.) -graphs, the limit (Formula presented.) exists, and “jumps” from 0 to (Formula presented.), that is, it never takes on values in the interval (Formula presented.). Moreover, we characterize which (Formula presented.) -graphs (Formula presented.) have (Formula presented.). 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.