Set system intersections can typically be blown up

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 $i \neq j$, if $T_i \in \mathcal{F}_i$ and $T_j \in \mathcal{F}_j$, then $T_i \cap T_j=S_i \cap S_j$.Comment: comments welcom

Monochromatic Sums and Products of Polynomials

We show that the pattern $\{x,x+y,xy\}$ is partition regular over the space of formal polynomials of degree at least one. This generalizes and provides a new proof of Moreira's celebrated result on the partition regularity of $\{x,x+y,xy\}$ over $\mathbb{N}$

Hindman's Conjecture over the Rationals

We show that the pattern $\sum_{i \in S}x_i, \prod_{i \in S}x_i$ is partition regular over the rationals, settling Hindman's conjecture over $\mathbb{Q}$

Improved bounds for the sunflower lemma

A sunflower with $r$ petals is a collection of $r$ sets so that the intersection of each pair is equal to the intersection of all. Erd\H{o}s and Rado proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$, with at least about $w^w$ sets, must contain a sunflower. The famous sunflower conjecture is that the bound on the number of sets can be improved to $c^w$ for some constant $c$. In this paper, we improve the bound to about $(\log w)^w$. In fact, we prove the result for a robust notion of sunflowers, for which the bound we obtain is tight up to lower order terms.Comment: Revised preprint, added sections on applications and rainbow sunflower

Discrepancy Minimization via a Self-Balancing Walk

We study discrepancy minimization for vectors in $\mathbb{R}^n$ under various settings. The main result is the analysis of a new simple random process in multiple dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for several problems in online vector balancing posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as linear time algorithms of logarithmic bounds for the Komlós conjecture.Non UBCUnreviewedAuthor affiliation: Princeton UniversityGraduat