New lower bounds for Ramsey numbers of graphs and hypergraphs

Let G be an r-uniform hypergraph. The multicolor Ramsey number rk(G) is the minimum n such that every k-coloring of the edges of the complete r-uniform hypergraph K n yields a monochromatic copy of G. Improving slightly upon results from [1], we prove that tk + 1 ≤ rk(K2,t+1) ≤ tk + k + 2, where the lower bound holds when t and k are both powers of a prime p. When t = 1, we improve the lower bound by 1, proving that rk(C4) ≥ k + 2 for any prime power k. This extends the result of [11] which proves the same bound when k is an odd prime power. These results are generalized to hypergraphs in the following sense. Fix integers r, s, t ≥ 2. Let H(s, t) be the complete r-partite r-graph with r − 2 parts of size 1, one part of size s, and one part of size t (note that H(s, t) = Ks,t). We prove tk − k + 1 ≤ rk(H(2, t+ 1)) ≤ tk + k + r, where the lower bound holds when t and k are both powers of a prime p; and k − ks−1 ≤ rk(H(s, t)) ≤ O(k), for fixed t, s ≥ 2, t > (s− 1)!; rk(H(3, 3)) = (1 + o(1))k, where the lower bound holds when k is a prime power. Some of our lower bounds are special cases of a family of more general hypergraph constructions obtained by algebraic methods. We describe these, thereby extending results of [12] about graphs. ∗Department of Mathematical Sciences, University of Delaware, Newark, DE19716, and DIMACS; research supported by NSA Grant No. MSPF–99G-170. †School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160; research supported in part by NSF Grant No. DMS–9970325. Current address: Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399 2000 Mathematics Subject Classification: 05B25, 05C35, 05C55 05C65, 05D05, 05D10, 11T99