Extremal and Ramsey results on graph blowups Fox, Jacob Luo, Sammy Wigderson, Yuval Recently, Souza introduced blowup Ramsey numbers as a gener- alization of bipartite Ramsey numbers. For graphs G and H, say G r −→ 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] r −→ 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. International Press of Boston 2021 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/22161 Fox J, Luo S, Wigderson Y. Extremal and Ramsey results on graph blowups. <i>Journal of Combinatorics</i>. 2021;12(1):1-15. doi:<a href="https://doi.org/10.4310/joc.2021.v12.n1.a1">10.4310/joc.2021.v12.n1.a1</a> eng info:eu-repo/semantics/altIdentifier/doi/10.4310/joc.2021.v12.n1.a1 info:eu-repo/semantics/altIdentifier/issn/2156-3527 info:eu-repo/semantics/altIdentifier/e-issn/2150-959X info:eu-repo/semantics/altIdentifier/arxiv/1912.08328 info:eu-repo/semantics/openAccess