A proof of the Elliott-R\"{o}dl conjecture on hypertrees in Steiner triple systems

TL;DR

This paper proves the Elliott-R"{o}dl conjecture, demonstrating that large Steiner triple systems contain all sufficiently small hypertrees, confirming a key hypothesis in hypergraph theory.

Contribution

The paper provides a proof of the Elliott-R"{o}dl conjecture, establishing the universal presence of small hypertrees in large Steiner triple systems.

Findings

Confirmed the conjecture for all sufficiently large systems.

Hypertrees with up to (1-μ)n vertices are contained in large Steiner triple systems.

Advances understanding of hypergraph embedding properties.

Abstract

Hypertrees are linear hypergraphs where every two vertices are connected by a unique path. Elliott and R\"{o}dl conjectured that for any given $\mu>0$, there exists $n_0$ such that the following holds. Every $n$-vertex Steiner triple system contains all hypertrees with at most $(1-\mu)n$ vertices whenever $n\geq n_0$. We prove this conjecture.