Extremal And Ramsey Results On Graph Blowups

Souza introduced blowup Ramsey numbers as a generalization of bipartite Ramsey numbers. For graphs G and H, say G (sic) H if every r-edge-coloring of G contains a monochromatic copy of H. Let H[t] denote the t-blowup of H. Then the blowup Ramsey number of G, H, r, and t is defined as the minimum n such that G[n] (sic) H[t]. Souza proved upper and lower bounds on n that are exponential in t, and conjectured that the exponential constant does not depend on G. We prove that the dependence on G in the exponential constant is indeed unnecessary, but conjecture that some dependence on G is unavoidable.An important step in both Souza's proof and ours is a theorem of Nikiforov, which says that if a graph contains a constant fraction of the possible copies of H, then it contains a blowup of H of logarithmic size. We also provide a new proof of this theorem with a better quantitative dependence.