## Probability Seminar

Abstract:

A distinct covering system of congruences is a list of congruences

\[

a_i \bmod m_i, \qquad i = 1, 2, ..., k

\]

whose union is the integers. Erdös asked if the least modulus $m_1$ 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.