Every Steiner triple system contains an almost spanning d-ary hypertree

TL;DR

This paper proves that large Steiner triple systems contain all perfect d-ary hypertrees on a significant subset of vertices, advancing understanding of their structural richness.

Contribution

It establishes that for sufficiently large systems, every perfect d-ary hypertree is contained, confirming a special case of a broader conjecture.

Findings

Confirmed the conjecture for perfect d-ary hypertrees

Demonstrated the existence of almost spanning hypertrees in large Steiner systems

Provided partial progress towards the general conjecture

Abstract

In this paper we make a partial progress on the following conjecture: for every $\mu>0$ and large enough $n$, every Steiner triple system $S$ on at least $(1+\mu)n$ vertices contains every hypertree $T$ on $n$ vertices. We prove that the conjecture holds if $T$ is a perfect $d$-ary hypertree.