There are two natural complexity measures associated with DNFs: their size, which is the number of clauses; and their width, which is the maximal number of variables in a clause. It is a folklore result that DNFs of small size can be approximated by DNFs of small width (logarithmic in the size). The other direction is much less clear.

Gopalan, Meka and Reingold [Computational Complexity 2013] showed that the other direction -- DNF sparsification -- holds as well. Any DNF of width $w$ can be approximated to within error $\epsilon$ by a DNF of size $(w\log(1/\epsilon))^{O(w)}$. Our main interest in this work is the dependence on the width $w$. The same dependence of $w^w$ appears in several other open problems in combinatorics and complexity, such as the Erd\H{o}s-Rado sunflower conjecture and Mansour's conjecture. In fact, there are deep connections between these three problems. Our main result is DNF compression with an improved dependence on the width, which overcomes the $w^w$ barrier. Concretely, we show that any DNF of width $w$ can be approximated to within error $\epsilon$ by a DNF of size $(1/\epsilon)^{O(w)}$.

The proof centers around a new object which we call the DNF index function. Given a DNF, the DNF index function outputs for an input the first clause that satisfies it (if one exists). Our proof has two parts: a combinatorial part, where we exhibit a switching lemma for the DNF index function; and an analytic part, where we use the switching lemma to bound the noise sensitivity of the DNF index function, and then use it to obtain our DNF compression result.