The tripartite Ramsey number for trees

TL;DR

This paper proves a new result in Ramsey theory, showing that in any two-colouring of a tripartite complete graph, one colour contains all trees of certain size and degree constraints, confirming a longstanding conjecture.

Contribution

It establishes the tripartite Ramsey number for trees, confirming Schelp's conjecture for large n and specific tree parameters.

Findings

One colour contains all trees of size less than approximately 1.5n.

The result holds for trees with maximum degree up to n^c.

The theorem applies for sufficiently large n and small epsilon.

Abstract

We prove that for all epsilon>0 there are c>0 and n_0 such that for all n>n_0 the following holds. For any two-colouring of the edges of $K_{n,n,n}$ one colour contains copies of all trees T of order t<(3-epsilon)n/2 and with maximum degree at most n^c. This confirms a conjecture of Schelp.