On Extremal k-Graphs Without Repeated Copies of 2-Intersecting Edges

TL;DR

This paper determines the maximum size of k-uniform hypergraphs with at most one pair of 2-intersecting edges for k=3 and k=4, extending extremal hypergraph theory with new exact results.

Contribution

It provides the first complete solutions for k=3 and nearly complete solutions for k=4 regarding extremal hypergraphs with limited 2-intersecting edges.

Findings

Complete solution for k=3 case.

Almost complete solution for k=4 case with eleven exceptions.

Extends extremal hypergraph results to new hypergraph families.

Abstract

The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turan numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for k in {3,4}. We give a complete solution when k=3 and an almost complete solution (with eleven exceptions) when k=4.