Gallai-Colorings of Triples and 2-Factors of B[subscript 3]

Abstract

A coloring of the edges of the r-uniform complete hypergraph is a G[subscript r]-coloring if there is no rainbow simplex; that is, every set of r + l vertices contains two edges of the same color. The notion extends G[subscript 2]-colorings which are often called Gallai-colorings and originates from a seminal paper of Gallai. One well-known property of G[subscript 2]-colorings is that at least one color class has a spanning tree. J. Lehel and the senior author observed that this property does not hold for G[subscript r]-colorings and proposed to study f[subscript r](n), the size of the largest monochromatic component which can be found in every G[subscript r]-coloring of K[r over n], the complete r-uniform hypergraph. The previous remark says that f[subscript 2](n) = n, and in this note, we address the case r = 3. We prove that [(n + 3)/2] ≤ f[subscript 3](n) ≤ [4n/5], and this determines f[subscript 3](n) for n < 7. We also prove that f[subscript 3](7) = 6 by excluding certain 2-factors from the middle layer of the Boolean lattice on seven elements.