I am looking to buy a saucepan. The 3 sizes available are 16cm, 18cm and 20cm.  I am tempted to get the largest as I'd rather have too much space than not enough, plus (I'm not sure if this is relevant), but  I think that with a distribution of liquid over a wider surface would help cook the recipe more evenly and faster.  Is it better to have a larger saucepan, or are there times when a smaller saucepan is better?  I would like to use it to cook soup, potatoes, carrots, spaghetti, etc. Mainly for 1-2 people. 
I am not sure if you want a pot (sides at least 10 cm high) or a pan (like a frying pan or saute pan.) I'll answer for pan since that's what you said. For a pot, it's less critical assuming it holds all the food. You can put a shallow amount of water in a big pot or a deep amount of water in a smaller pot, which may then boil over. You might as well get the bigger one if you're only getting one, unless it's bigger than the stove ring you intend to put it on.  On the matter of a pan, it's impossible to give a specific answer without knowing how many people you are cooking for. A pan that's right for a one-pot meal for 6 people is far too big for say frying one egg in butter - you'll need to use way more butter and you'll have some burning in the parts of the pan that don't have any egg on them.  The disadvantage of a too-small pan is that fried food will steam instead of frying due to crowding, or that what you're cooking will spill over the edges making much more mess in your kitchen. The disadvantage of a too-large pan is that it's heavy, takes up a lot of room, and possibly scorches and burns in the empty spots, or reduces too fast if it's ful of liquid, again with possibly scorching.  I like to have one pan that just holds two eggs, for frying one or two eggs, and one that's about the size of a dinner plate, for everything else (omelettes for two, sauteed veg for 4, the filling for shepherds pie that feeds 8, poaching fish, etc.) I also have one super large pan that I keep in the pantry and haul out for things like frying 6 hamburger patties at once. But I could get by without that one.  If you're cooking for one and can only get one pan at the moment, don't get it larger than a dinner plate. Then pay attention to whether it is more often too big or too small, and keep that in mind when the time comes to buy a second one. 