How to protect large empty space?

At move 41, black is trying to take part of bottom right corner. What is the stone efficient way to avoid that?

Don’t allow your opponent to form two eyes.
If you play moves that make it impossible for your opponent to form a basis along the sides, you are almost there.
Avoid if possible attachments.

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Although black is still dead after your move 42 (you could pass and black is still dead), it reveals a mistake in thinking. Your 42 doesn’t stop black making eyespace along the side/corner, but instead stops him walking into your huge wall which he doesn’t want to do anyway as there’s no eyes that way. e2 would be a better idea, preventing black from making eyespace along the side.

The general principles to stopping such invasions from living are:

  1. Defend your weaknesses that would allow opponent to capture some of your surrounding walls and live (not applicable here)
  2. Play from outside to prevent them making enough space to make 2 eyes. These should generally be non-contact moves, as contact moves give opponent sente moves in a local battle.
  3. Only once opponent is safely surrounded and threatening to split the eyespace into 2 eyes (e.g a straight 3) do you play a vital point inside.

A lot of beginners make the mistake of trying to take liberties to directly capture the invading stones rather than do point 1 or 2, and do point 3 too early, your move 50 is an example, j2 from outside is better (though both do kill).

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I need help too.
My opponents overran the territories I had managed to build, and I had no idea how to defend them.

Well, here is some politically incorrect advice.
If you don’t want immigrant stones to flood your territory, close your borders before they get a chance to do so.

Read my list and think about what applies to your game situation for every move they played inside your territory.

K5 was a big mistake, as pointed out by the AI.

There are many ways to tackle this, but I’d surmise by saying:

Your bottom group is strong, so no need to protect / attack from that side. Instead, attack or protect from the side that is weaker.

If you didn’t notice the cut at L9 was dangerous before the opponent played it, then make sure to look for those types of cuts. They are always something you need to be careful about, especially in endgame.

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