Set System Blowups
Ryan Alweiss · COMBINATORICA · 2025
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 $$i e j$$ , 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.