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To exclude an ingredient, add "-" before the ingredient name. For example, searching EYB recipes for carrots gives 126,000 recipes (using round numbers). Searching EYB recipes for "carrots -potatoes" (i.e., carrots, no potatoes) yields 105,000 recipes. "Carrots -potatoes -pasta" yields 95,000. Also remember which ingredients are included in the store-cupboard ingredients, so small quantities may not be indexed. 

Actually a better way to exclude ingredients is to use the filters. You select an ingredient category in the right "Only Show..." column then click the + to change it to a -. This is explained further in Help. As lkgrover said, you do need to be aware that some store-cupboard ingredients are not listed in small quantities so even if you exclude that ingredient e.g. flour, you may still get some recipes with small amounts of the ingredient.

A word search is not as precise a match as the category and ingredient tags because with recipes and ingredients the word does not always appear in the name e.g. pasta may be listed as ramen or soba or bean thread noodles etc. Whereas if you select the entire categories of Long pasta, Noodles and Pasta shapes then exclude them you will be cutting out all pasta and noodles ingredients rather than just ingredients that contain the word "pasta".

You also have to be aware that recipes for an ingredient rather than containing an ingredient will not be excluded if you exclude the ingredient filters. This isn't an issue for most ingredients but in the case of pasta, you would want to enter -pasta in the word search to also exclude all recipes with pasta in the name. So you can see how that search would look, here are the search results for the Library.

I was once asked to find recipes that exclude onions and garlic as ingredients for a friend due to his dietary needs.
It seemed to me impossible since these are store cupboard ingredients with a minimum quantity limit.
I don't see a way to substitute those ingredients or omit them like sugar or flour. How do you feel about it?

Braco777 - I'm not clear what your question is. Yes, you cannot exclude onions or garlic as ingredients and get a list of recipes without them since onions and garlic are considered store-cupboard ingredients so will not be listed unless they are a major ingredient in the recipe e.g. French onion soup or garlic bread.