Positive Co‐Degree Density of Hypergraphs

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