Embedding Hypertrees into Steiner Triple Systems

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 $m\ge (1+\mu )n$? We show the answer is positive for a class of hypertrees and conjecture that the answer is always positive.

23 pages, 6 figures