Set System Blowups

Abstract

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 $i\ne j$, 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.

many edits made, to appear in Combinatorica