Embedding Hypertrees into Steiner Triple Systems

TL;DR

This paper investigates whether large Steiner triple systems necessarily contain certain hypertrees as subgraphs, proving it for some classes and conjecturing it generally.

Contribution

It establishes the presence of specific hypertrees in sufficiently large Steiner triple systems and proposes a conjecture for the universal case.

Findings

Confirmed the embedding of some hypertrees in Steiner systems

Proved the main result for a particular class of hypertrees

Conjectured the general embedding property for all hypertrees

Abstract

In this paper we are interested in the following question: Given an arbitrary Steiner triple system $S$ on $m$ vertices and any 3-uniform hypertree $T$ on $n$ vertices, is it necessary that $S$ contains $T$ as a subgraph provided $m \geq (1+\mu)n$? We show the answer is positive for a class of hypertrees and conjecture that the answer is always positive.