Extremal Problems on Generalized Directed Hypergraphs

TL;DR

This paper extends extremal combinatorics concepts like Turan density to a broad class of uniform, simple directed hypergraphs, establishing fundamental properties and relationships among jumps and densities.

Contribution

It introduces a generalized framework for directed hypergraphs and extends key extremal results, including supersaturation and blowup density invariance, within this new setting.

Findings

Supersaturation holds for generalized directed hypergraphs.

Blowup of a GDH has the same Turan density as the original.

Degenerate GDHs are characterized by being contained in a blowup of a single edge.

Abstract

In this paper we define a class of combinatorial structures the instances of which can each be thought of as a model of directed hypergraphs in some way. Each of these models is uniform in that all edges have the same internal structure, and each is simple in that no loops or multiedges are allowed. We generalize the concepts of Turan density, blowup density, and jumps to this class and show that many basic extremal results extend naturally in this new setting. In particular, we show that supersaturation holds, the blowup of a generalized directed hypergraph (GDH) has the same Turan density as the GDH itself, and degenerate GDHs (those with Turan density zero) can be characterized as being contained in a blowup of a single edge. Additionally, we show how the set of jumps from one kind of GDH relates to the set of jumps of another. Since r-uniform hypergraphs are an instance of the defined class, then we are able to derive many particular instances of jumps and nonjumps for GDHs in general based on known results.