A note on embedding hypertrees

TL;DR

This paper proves that in r-uniform hypergraphs, a chromatic number greater than the number of edges t guarantees the embedding of any r-tree with t edges, resolving a question about the dependence on r.

Contribution

It establishes the tight bound that a chromatic number exceeding t suffices for embedding any r-tree with t edges, removing the previous dependence on r.

Findings

Proves hi > t is sufficient for embedding any r-tree with t edges.

Resolves a question about the necessity of the dependence on r.

Provides a tight bound for embedding r-trees in hypergraphs.

Abstract

A classical result from graph theory is that every graph with chromatic number \chi > t contains a subgraph with all degrees at least t, and therefore contains a copy of every t-edge tree. Bohman, Frieze, and Mubayi recently posed this problem for r-uniform hypergraphs. An r-tree is an r-uniform hypergraph with no pair of edges intersecting in more than one vertex, and no sequence of distinct vertices and edges (v_1, e_1, ..., v_k, e_k) with all e_i \ni {v_i, v_{i+1}}, where we take v_{k+1} to be v_1. Bohman, Frieze, and Mubayi proved that \chi > 2rt is sufficient to embed every r-tree with t edges, and asked whether the dependence on r was necessary. In this note, we completely solve their problem, proving the tight result that \chi > t is sufficient to embed any r-tree with t edges.