[Theory] a seminar of interest

[Alexander Razborov]

The probability seminar this week, given by Bob Hough from Stony Brook, may also be of interest to some on this list. The talk will be at 2:30 p.m. on Friday, Jan. 28 in Eckhart 202, and will be simultaneously broadcast on zoom, using this link:

https://uchicago.zoom.us/j/5478153078?pwd=Y09DRndpVVE2V0tycWhlMUFvTUVmdz09 

Title: Covering systems of congruences 
Abstract: A distinct covering system of congruences is a list of congruences ai ≅ mi, for i = 1, 2, ..., k whose union is the integers. Erdős asked if the least modulus m1 of a distinct covering system of congruences can be arbitrarily large (the minimum modulus problem for covering systems, $1000) and if there exist distinct covering systems of congruences all of whose moduli are odd (the odd problem for covering systems, $25). I'll discuss my proof of a negative answer to the minimum modulus problem, and a quantitative refinement with Pace Nielsen that proves that any distinct covering system of congruences has a modulus divisible by either 2 or 3. The proofs use the probabilistic method. Time permitting, I may briefly discuss a reformulation of our method due to Balister, Bollobás, Morris, Sahasrabudhe and Tiba which solves a conjecture of Shinzel (any distinct covering system of congruences has one modulus that divides another) and gives a negative answer to the square-free version of the odd problem.
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