## Mathematics, Department of

## Date of this Version

10-23-2021

## Citation

Published (2023) SIAM Journal on Discrete Mathematics, 37 (3), pp. 1818-1841. DOI: 10.1137/21M1451427

## Abstract

Motivated by the Beck-Fiala conjecture, we study the discrepancy problem in two related models of random hypergraphs on *n* vertices and *m* edges. In the first (*edge-independent*) model, a random hypergraph *H*_{1} is constructed by fixing a parameter *p* and allowing each of the *n* vertices to join each of the *m* edges independently with probability *p*. In the parameter range in which *pn* ⟶ ∞ and pm ⟶ ∞, we show that with high probability (*w.h.p*.) *H*_{1} has discrepancy at least Ω(2^{-n/m} √*pn*) when *m* = *O*(*n*), and at least Ω( √*pn*logү) when *m* ≽ *n*, where ү= min{*m/n*; *pn*}. In the second (*edge-dependent)* model, *d* is fixed and each vertex of *H*_{2} independently joins exactly d edges uniformly at random. We obtain analogous results for this model by generalizing the techniques used for the edge-independent model with *p* = *d*/*m*. Namely, for *d* ⟶ ∞ and *dn*/*m *⟶ ∞, we prove that* w.h.p*. *H*_{2} has discrepancy at least Ω(2^{-n/m} √*dn*/*m*) when *m* = *O*(*n*), and at least Ω( √(*dn/m*) logү) when *m *≽ *n*, where ү= min{*m*/*n*; *dn*/*m*}. Furthermore, we obtain nearly matching asymptotic upper bounds on the discrepancy in both models (when *p* = *d*/*m*), in the dense regime of *m *≽ *n*. Specifically, we apply the partial colouring lemma of Lovett and Meka to show that *w.h.p*. *H*_{1} and *H*_{2} each have discrepancy *O*( √*dn*/*m* log(*m*/*n*)), provided d ⟶ ∞, dn=m ⟶ ∞ and *m *≽ *n*. This result is algorithmic, and together with the work of Bansal and Meka characterizes how the discrepancy of each random hypergraph model transitions from Θ( √*d*) to *o*( √*d*) as m varies from *m* = Θ(*n*) to *m *≽ *n*.

## Comments

Used by permission.