Finding large rainbow trees in colourings of Kn,n - Matthes, Julian (The electronic journal of combinatorics, 2023)
A subgraph of an edge-coloured graph is called rainbow if all of its edges have distinct colours. An edge-colouring is called locally k-bounded if each vertex is incident with at most k edges of the same colour. Recently, Montgomery, Pokrovskiy and Sudakov showed that for large n, a certain locally 2-bounded edge-colouring of the complete graph K2n+1 contains a rainbow copy of any tree with n edges, thereby resolving a long-standing conjecture by Ringel: For large n, K2n+1 can be decomposed into copies of any tree with n edges. In this paper, we employ their methods to show that any locally k-bounded edge-colouring of the complete bipartite graph Kn,n contains a rainbow copy of any tree T with (1 - o(1))n/k edges. We show that this implies that every tree with n edges packs at least n times into Kn+o(1),n+o(1). We conjecture that for large n, Kn,n can be decomposed into n copies of any tree with n edges.