Extremal and Ramsey results on graph blowups

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