For fixed r ? 2 , an r -graph ? G = ( V , E ) is an r -uniform hypergraph with vertex set V and edge set E ? ( V r ) . For k ? r ? 2 , let K k denote the complete r -graph on k vertices and let K k ? 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 ? K k or F ? K k ? , then this fails to be true. On the other hand, a result of Abramson and Harrington (1978) and Nesetřil and Rodl (1977) implies that F = K k , as well as F = K k ? , has the Ramsey property.The proof of the Induced Ramsey Theorem was based on a partite lemma and partite construction as shown in Nesetřil and Rodl (1977) and Nesetřil and Rodl (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.
['Vindya Bhat', 'Vojtěch Rödl']
A short proof of the induced Ramsey Theorem for hypergraphs