Triangle-free triple systems

TL;DR

This paper systematically studies fifteen extremal problems related to triangle-free configurations in 3-uniform hypergraphs, providing exact or asymptotic solutions and characterizations of extremal structures.

Contribution

It offers a comprehensive analysis of all configurations avoiding certain triangle formations, solving new cases and characterizing extremal hypergraphs.

Findings

Several extremal problems solved exactly or asymptotically

New characterizations of extremal hypergraph constructions

Extension of known results to all triangle-avoidance configurations

Abstract

There are four non-isomorphic configurations of triples that can form a triangle in a $3$-uniform hypergraph. Forbidding different combinations of these four configurations, fifteen extremal problems can be defined, several of which already appeared in the literature in some different context. Here we systematically study all of these problems solving the new cases exactly or asymptotically. In many cases we also characterize the extremal constructions.