Mixelv

SELECT `c`.`name`
    FROM `cabinets` AS `c`
    WHERE `c`.`id` NOT IN (
        SELECT DISTINCT `b`.`cabinet_id`
            FROM `busy` AS `b` 
            WHERE `b`.`date_start` BETWEEN {$s} AND {$e} 
                OR {$s} BETWEEN `b`.`date_start` AND `b`.`date_end`
    )