Set System Blowups

We prove that given a constant $k\ge 2$$k \ge 2$ and a large set system $\mathcal{F}$$\mathcal {F}$ of sets of size at most w, a typical k-tuple of sets $(S_1,\cdots ,S_k)$$(S_1, \cdots, S_k)$ from $\mathcal{F}$$\mathcal {F}$ can be “blown up” in the following sense: for each $1\le i\le k$$1 \le i \le k$, we can find a large subfamily ${\mathcal{F}}_i$$\mathcal {F}_i$ containing $S_i$$S_i$ so that for $i\ne j$$i \ne j$, if $T_i\in {\mathcal{F}}_i$$T_i \in \mathcal {F}_i$ and $T_j\in {\mathcal{F}}_j$$T_j \in \mathcal {F}_j$, then $T_i\cap T_j=S_i\cap S_j$$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.