Room for a ten city empire ? Nope. Not a given. Not on a huge map at least. Ai civ's start ten hexes apart.
I'm currently playing on a normal-sized map with a normal amount of civs, where I'd expect less room per civ than on a huge map, and I'm playing the lakes map script, which has a lot of land and very little water.
I have (turn 68) 7 cities built right now, and 12 pins of planned cities, and based on the distance of the AIs and positioning of city states, I'd be surprised if I lost even a
single one of those pins to an AI city. That's 19 cities, with possibly more that I don't currently have planned out. The closest AI capital to my capital is about 25 tiles away.
If you don't have room, play a different map script.
...In fact, you know what, I'm checking my recent games.
In my Australia game, I'm in the Industrial Era, and I'm playing a Highlands map. Again, standard map size, standard number of civs. Note that this game I was using the Better Balanced Starts mod, which adjusts the mechanics a bit. I have 15 cities, with 7 more planned and no wars fought (except against the Barbarians, lmao). The closest AI capital is 16 tiles away from my capital, but it should be noted that in one direction I have more like 25 tiles room to exapnd.
In my Babylon game, I'm in the Future Era in my last save. Standard map size, standard number of civs, Seven Seas map script. I have 15 cities, though admittedly 5 of those were flipped. One of those five flips happened because the AI forward settled me, and if it hadn't flipped I would have conquered it because it was easy pickings, and then possibly razed it because that way I could've fit in another city that now didn't fit. It should be noted that this game I was extremely slow with expanding; I only got my second city down in like turn 42 or something. Notably, the closest capital to mine was only 11 tiles away, though the next closest was 16 tiles away. For all accounts and purposes, I was playing a tall game, and had I wanted to, I could've probably settled two or three more cities before the AI got to certain locations.
In my One-City Challenge Babylon game, I won't bother counting the cities, obviously. I was playing standard map size, standard number of civs, and a pangea map. The closest capital to mine was 11 tiles away, the next 16 tiles. This game would've been rather cramped if I hadn't been playing OCC, I think, and I'd still have been able to get about 8 cities in.
In my Phoenicia game, again standard map size and number of civs, I was playing an archipilago map - the lowest land area possible. This is a domination game, which of course changes things a bit, however I peacefully settled 9 cities that I'd have gotten no matter what, there are another 3 that I might have lost to AI expansion if I hadn't fought that AI (they declared war on me by the way) and there are another 4 city locations that I'd have settled before the AI got there if I hadn't been focused on war, for a total of at least 13 cities without war. The closest AI capital is 13 tiles away from mine.
In my Cree game, I was again using the Seven Seas script and Better Balanced Starts mod. In the last save, I'm in the Industrial Era (I had a very early diplo victory), with zero wars fought all game, and I have 16 cities. Three of these were flipped, however, one was an absolutely braindead settle by the AI, and would have been three cities had I had the opportunity to settle the area myself. There are also still three good spots for cities left that I never bothered to settle, and two or three spots where an AI has settled that I could have gotten myself had I wanted to. The closest AI capital is 16 tiles away from mine.
In my Vietnam game, which I kind of abandoned, I'm in the early Industrial Era. I'm not entirely sure of the map script because it's too long ago, but I suspect it's Continents and Islands based on what I'm seeing. I peacefully settled 14 cities, with room for 1 more, and that's while I got forward settled by the AI, losing me 3-4 cities I was planning on settling. The closest AI capital is 15 tiles away.
How many more examples do you want?
EDIT: For completeness' sake, these 7 games were my last seven games that I have at least one save of, with the exception of:
-Two games where I played with a mod that unlocks every single unique ability in the game, making me incredibly overpowered.
-One game where I played with more than the default number of civs, which I abandoned relatively early on, in part because of how crowded it was with that many civs, but I also cannot use it for this post because I don't know where the AI capitals are.
-Two starts I saved because they looked interesting when I was rolling starts, but I haven't played with these yet so I don't know how far the AI is.