Usually when the mashing process is over and the liquid is transferred to the fermenter there's a head-space of air. Lately I've made smaller batches of approximately 5 liters (a huge inefficiency, i know) but my bucket has 20 liters of volume. So, does this enormous head-space places a greater chance of contamination? If so, how sensitive is this process (like does the amount of air increases exponentially the chances of contamination)? Tks!
It's not contamination you should be worried about, but oxidation. More headspace equals more oxygen in contact with your wort. This shouldn't be a major concern during primary fermentation, because the CO2 molecules generated by the yeast purge the oxygen from your fermenter (through the airlock). However, I don't have any experience or data to say just how much CO2 is created during a typical fermentation, and 15 liters seems like a lot of space to purge.
There is a very good article on MoreBeer that has some data showing how an increase in headspace dramatically reduces the time to staling (production of undesirable off flavors). The article, and my experience, indicate that oxidation is more of a problem post fermentation, since there is no longer a production of CO2 to drive oxygen from your fermenter.
So, while you should be able to rest easy during primary fermentation -- even with your large amount of headspace -- you should definitely do what you can to reduce headspace to the absolute minimum in secondary fermentation (if you do that) and in bottling.
I would, as a matter of good practice, get containers that more closely match your batch sizes. I like to use carboys, since they have narrow necks. I use a 6 gallon (23 liter) carboy for primary and fill it to within 1/2 gallon of the top. I then transfer to a smaller 5 gallon (19 liter) carboy for secondary and try to have no headspace at all. Other brewers may have different experience and recommendations.
At the beginning of the process, it is not sensitive, you're okay.
Initially, the wort needs oxygen for the yeast to reproduce. As it eats up the oxygen, the yeast will start producing CO2, which is heavier than typical air. CO2 will provide a barrier to any leftover oxygen in the headspace. Just don't slosh your fermentor around.
A bigger concern regarding oxygen headspace is if you transfer to a secondary. You'll want minimal headspace here.