How much exactly food is generally required to trigger the food allergy?

Even tiny bites . Depending on how allergic a person is to a food even a tiny bite can cause anaphylaxis. For less severe or non life-threatening allergies (like eczema) more is often required, but it varies with each patient.
Tiny amounts! If you are truly food allergic, minute amounts of a food protein can cause a serious reaction. That is why some airlines quit serving peanuts because the circulating peanut dust can cause a reaction. Foods now also carry a warning that they are manufactured in plants where peanuts, or nuts, or other food allergens, are also produced.