Extremal Numbers for 2 to 1 Directed Hypergraphs with Two Edges Part I: The Nondegenerate Cases

TL;DR

This paper determines the maximum number of edges in 2 to 1 directed hypergraphs with two edges that avoid certain configurations, focusing on nondegenerate cases where extremal numbers grow cubically with the number of vertices.

Contribution

It explicitly calculates the extremal numbers for three specific nondegenerate 2 to 1 directed hypergraphs with two edges, filling a gap in the understanding of these structures.

Findings

Standard and oriented extremal numbers are cubic in n for these three hypergraphs.

Identifies exactly which 2 to 1 hypergraphs with two edges have cubic extremal numbers.

Complements previous work by classifying these specific hypergraphs' extremal properties.

Abstract

Let a 2 to 1 directed hypergraph be a 3-uniform hypergraph where every edge has two tail vertices and one head vertex. For any such directed hypergraph F let the nth extremal number of F be the maximum number of edges that any directed hypergraph on n vertices can have without containing a copy of F. There are actually two versions of this problem: the standard version where every triple of vertices is allowed to have up to all three possible directed edges and the oriented version where each triple can have at most one directed edge. In this paper, we determine the standard extremal numbers and the oriented extremal numbers for three different directed hypergraphs. Each has exactly two edges, and of the seven (nontrivial) 2 to 1 graphs with exactly two edges, these are the only three with extremal numbers that are cubic in n. The standard and oriented extremal numbers for the other four directed hypergraphs with two edges are determined in a companion paper.