On $3$-uniform hypergraphs without linear cycles

TL;DR

This paper investigates 3-uniform hypergraphs without linear cycles, establishing that such hypergraphs must have vertices of limited degree and large independent sets, revealing structural properties of these hypergraphs.

Contribution

It proves that 3-uniform hypergraphs without linear cycles contain vertices of degree at most two and have large independent sets, advancing understanding of their structure.

Findings

Existence of vertices with strong degree at most two

Presence of independent sets of size at least 2|V(H)|/5

Structural constraints on hypergraphs without linear cycles

Abstract

We explore properties of $3$-uniform hypergraphs $H$ without linear cycles. Our main results are that these hypergraphs must contain a vertex of strong degree at most two and must have independent sets of size at least ${2|V(H)|\over 5}$.