On an Erdos-type conjecture on F₂x

In 1970, R. B. Crittenden and C. L. Vanden EyndenCV1970 proved that any system of n congruence classes covering positive integers 1, 2,. . . 2ⁿ covers all integers, which was originally conjectured by P. ErdosErdos1962 in 1962, who first proved a weaker version of this conjecture, replacing 2ⁿ with C (n), where C (n) is a constant determined by n. The constant 2ⁿ here cannot be improved, since one can find n congruences covering any given 2ⁿ consecutive integers with any one given exception among them exactly. In 2019, P. Balister, B. Bollob\'as, R. Morris, J. Sahasrabudhe and M. TibaBBST2020 gave a simpler proof of this conjecture also by the proof by contradiction, including constructing polymials using parameters from the counter-example congruences.