A Note on Embedding Hypertrees

A classical result from graph theory is that every graph with chromatic number χ > 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 runiform hypergraph with no pair of edges intersecting in more than one vertex, and no sequence of distinct vertices and edges (v1, e1, . . . , vk, ek) with all ei ∋ {vi, vi+1}, where we take vk+1 to be v1. Bohman, Frieze, and Mubayi proved that χ > 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 χ > t is sufficient to embed any r-tree with t edges.