Two Ramsey problems in blowups of graphs

Given graphs G and H, we say G→rH if every r-colouring of the edges of G contains a monochromatic copy of H. Let Ht denote the t-blowup of H. The blowup Ramsey number B(G→rH;t) is the minimum n such that Gn→rHt. Fox, Luo and Wigderson refined an upper bound of Souza, showing that, given G, H and r such that G→rH, there exist constants a=a(G,H,r) and b=b(H,r) such that for all t∈N, B(G→rH;t)≤abt. They conjectured that there exist some graphs H for which the constant a depending on G is necessary. We prove this conjecture by showing that the statement is true in the case of H being 3-chromatically connected, which in particular includes triangles. On the other hand, perhaps surprisingly, we show that for forests F, there exists an upper bound for B(G→rF;t) which is independent of G. Second, we show that for any r,t∈N, any sufficiently large r-edge coloured complete graph on n vertices with Ω(n2−1/t) edges in each colour contains a member from a certain finite family Ftr of r-edge coloured complete graphs. This answers a conjecture of Bowen, Hansberg, Montejano and Müyesser.