On an Erd\H{o}s-type conjecture on $\mathbb{F}_q[x]$

TL;DR

This paper extends Erdős' conjecture from integers to polynomial rings over finite fields, proving that certain covering sets of cosets imply the entire polynomial ring is covered.

Contribution

It generalizes Erdős' conjecture to polynomial rings over finite fields and provides a proof using methods similar to those in earlier integer cases.

Findings

Sets of n cosets covering all polynomials of degree less than n cover the entire ring.

The proof adapts classical approaches to the polynomial ring setting.

Confirms a polynomial analogue of Erdős' conjecture.

Abstract

P. Erd\H{o}s 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. Bollob\'{a}s, 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 Erd\H{o}s' 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 $\mathbb{F}_q[x]$.