ON MULTICOLOR RAMSEY NUMBERS AND SUBSET COLORING OF HYPERGRAPHS

For n ≥ s > r ≥ 1 and k ≥ 2, write n → (s)r k if every hyperedge coloring with k colors of the complete r-uniform hypergraph on n vertices has a monochromatic subset of size s. Improving upon previous results by M. Axenovich, A. Gyárfás, H. Liu, and D. Mubayi [Discrete Math., 322 (2014), pp. 69-77] and P. Erdos, A. Hajnal, A. Máté, and R. Rado, [Combinatorial set theory: Partition Relations for Cardinals, Elsevier, Amsterdam, 1984] we show that if r ≥ 3 and n → (s)r k, then 2n → (s+1)r+1 k+3. This improves some of the known lower bounds on multicolor hypergraph Ramsey numbers. Given a hypergraph H = (V,E), we consider the Ramsey-like problem of coloring all r-subsets of V such that no hyperedge of size ≥ r + 1 is monochromatic. We provide upper and lower bounds on the number of colors necessary in terms of the chromatic number χ(H). In particular we show that this number is O(log(r-1)(rχ (H))+r), where logy is the log function applied y times.