Blowup Ramsey numbers

Abstract

We study a generalisation of the bipartite Ramsey numbers to blowups of graphs. For a graph G, denote the t-blowup of G by G[t]. We say that G is r-Ramsey for H, and write G→rH, if every r-colouring of the edges of G has a monochromatic copy of H. We show that if G→rH, then for all t, there exists n such that G[n]→rH[t]. In fact, we provide exponential lower and upper bounds for the minimum n with G[n]→rH[t], and conjecture an upper bound of the form ct, where c depends on H and r, but not on G. We also show that this conjecture holds for G(n,p) with high probability, above the threshold for the event G(n,p)→rH.