On an Erdős-type conjecture on $\mathbb{F}_q[x]$

Abstract

P. Erdős conjectured in 1962 that on the ring $\mathbb{Z}$, every set of $n$ congruence classes in $\mathbb{Z}$ that covers the first $2^n$ positive integers also covers the ring $\mathbb{Z}$. This conjecture was first confirmed in 1970 by R. B. Crittenden and C. L. Vanden Eynden. Later, in 2019, P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe, and M. Tiba provided a more transparent proof. In this paper, we follow the approach used by R. B. Crittenden and C. L. Vanden Eynden to prove the generalized Erdős' conjecture in the setting of polynomial rings over finite fields. We prove that every set of $n$ cosets of ideals in $\mathbb F_q[x]$ that covers all polynomials whose degree is less than $n$ covers the ring $\mathbb{F}_q[x]$.