A short proof of the induced Ramsey Theorem for hypergraphs

Abstract

For fixed r≥2, an r-graph G=(V,E) is an r-uniform hypergraph with vertex set V and edge set E⊆(Vr). For k≥r≥2, let Kk denote the complete r-graph on k vertices and let Kk¯ denote its complement, an independent set on k vertices.The Induced Ramsey Theorem states that for c,r≥2 and every r-graph G, there exists an r-graph H such that every c-coloring of the edges of H contains a monochromatic induced copy of G. A natural question to ask is what other subgraphs F can be partitioned and have a similar Ramsey property. One can show that if F≠Kk or F≠Kk¯, then this fails to be true. On the other hand, a result of Abramson and Harrington (1978) and Nešetřil and Rödl (1977) implies that F=Kk, as well as F=Kk¯, has the Ramsey property.The proof of the Induced Ramsey Theorem was based on a partite lemma and partite construction as shown in Nešetřil and Rödl (1977) and Nešetřil and Rödl (1987). In this note, we present a short proof of this result by eliminating the partite construction to show that the theorem is a direct consequence of the Hales–Jewett Theorem. In other words, we have reduced the proof of this result to one step, rather than two steps.