I have heard there is an injury and illness recordkeeping regulation. Does an employer that has 3 employees have to comply with this regulation?

Employers that had no more than 10 employees at any time during the previous year are not required to keep injury and illness records under Part 1904. All employers are required to report work-related incidents involving fatalities, amputations, in-patient hospitalizations, or the loss of an eye to OSHA. Employers are also required to maintain the recordkeeping forms if they are notified in writing by OSHA or the BLS to do so.