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.
