New Lower Bounds for Ramsey Numbers of Graphs and Hypergraphs

Abstract

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(r)n yields a monochromatic copy of G. Improving slightly upon results from M. Axenovich, Z. Füredi, and D. Mubayi (J. Combin. Theory Ser. B79 (2000), 66–86), we prove thattk2+1≤rk(K2,t+1)≤tk2+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)≥k2+2 for any prime power k. This extends the result of F. Lazebnik and A. J. Woldar (J. Combin. Theory Ser. B79 (2000), 172–176), 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(r)(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(2)(s,t)=Ks,t). We prove thattk2−k+1≤rk(H(r)(2,t+1))≤tk2+k+r,where the lower bound holds when t and k are both powers of a prime p and(1−o(1))ks≤rk(H(r)(s,t))≤O(ks),for fixedt,s≥2,t>(s−1)!,rk(H(r)(3,3))=(1+o(1))k3.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 F. Lazebnik and A. J. Woldar (J. Graph Theory, 2001) about graphs.