Set System Blowups

We prove that given a constant $k\ge 2$ and a large set system $\mathcal{F}$ of sets of size at most $w$, a typical $k$-tuple of sets $(S_1,\cdots ,S_k)$ from $\mathcal{F}$ can be ``blown up" in the following sense: for each $1\le i\le k$, we can find a large subfamily ${\mathcal{F}}_i$ containing $S_i$ so that for $i\mathrm{\ne }j$, if $T_i\in {\mathcal{F}}_i$ and $T_j\in {\mathcal{F}}_j$ , then $T_i\cap T_j=S_i\cap S_j$. We also show that the answer to the multicolor version of the sunflower conjecture is the same as the answer for the original, up to an exponential factor.