We study discrepancy minimization for vectors in ℝn under various settings. The main result is the analysis of a new simple random process in high dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for online vector balancing against oblivious adversaries, resolving several questions posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as a linear time algorithm for logarithmic bounds for the Komlós conjecture.

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. Erdős 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 (logw) 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.