Ramsey Properties of Random Hypergraphs

LetK(k)(n, p) be the randomk-uniform hypergraph obtained by independent inclusion of each of the[formula]k-tuples with probabilityp. For an arbitraryk-uniform hypergraphGand every integerrwe find the threshold for the property that everyr-coloring of the vertices ofK(k)(n, p) results in a monochromatic copy ofG. In the edge coloring case, which is of our main interest, we find the threshold only fork=3,r=2, andG=K34. In the proof we utilize a recent version of the hypergraph regularity lemma due to Frankl and Rِdl.