What are the product category codes used of the products?
select product_category_code from products
What is the ones used most frequently?
select product_category_code from products group by product_category_code order by count(*) desc limit 1
What about for the characteristic name instead?
select t3.characteristic_name from products as t1 join product_characteristics as t2 on t1.product_id = t2.product_id join characteristics as t3 on t2.characteristic_id = t3.characteristic_id group by t3.characteristic_name order by count(*) desc limit 1