dvoretsky

Understand the 5 or so things (given below). If any of them don't make sense, then set up a board and play some moves against yourself until you convince yourself it is true.
1) If you get opposition on the e file it doesn't win (because the black pawn controls e5)
2.1) If you get opposition on the f file then you win because black must always step towards the queenside and...
2.2) When there's a file in between, then whoever lands on the file first will lose opposition.
2.3) So now we know when there's a file in between, the defender's king is forced to switch from defending along a rank to defending along a file (you're "outflanking" them).
3) If you walk up the g file, black cannot keep you out by defending along the e file (your pawn controls e6).
Therefore putting a file in between and walking up the g file means black will always lose opposition, so your king will infiltrate.
Calling it a major line is a little confusing / pointless IMO. The key features are what I explained: you outflank in a scenario where the file (or rank) they switch to is unusable.

Complicated positions require a lot of calculation.
But also you make it easier on yourself by establishing win conditions (you can also call this finding key squares). You just imagine the kings to be wherever you want, and then check whether that's a win or draw. Once you discover 1 or more situations like this, then they'll guide your calculation.
I think the main thing to get from the example you're talking about is the idea that outflanking forces the defender to switch from file to rank or vice versa.
Understand the 5 or so things (given below). If any of them don't make sense, then set up a board and play some moves against yourself until you convince yourself it is true.
1) If you get opposition on the e file it doesn't win (because the black pawn controls e5)
2.1) If you get opposition on the f file then you win because black must always step towards the queenside and...
2.2) When there's a file in between, then whoever lands on the file first will lose opposition.
2.3) So now we know when there's a file in between, the defender's king is forced to switch from defending along a rank to defending along a file (you're "outflanking" them).
3) If you walk up the g file, black cannot keep you out by defending along the e file (your pawn controls e6).
Therefore putting a file in between and walking up the g file means black will always lose opposition, so your king will infiltrate.
Calling it a major line is a little confusing / pointless IMO. The key features are what I explained: you outflank in a scenario where the file (or rank) they switch to is unusable.
Thank you for your help. I've been sitting at a board for the last 30 minutes trying to understand the points you made. And while I see the points and the practice I'm sure will help me in the long run, I feel like maybe this particular manual is quite a bit over my playing level so I will find a more fundamental book and maybe come back to this book once I'm a little stronger.

Oh, yeah, I just saw your rating, this is a pretty advanced example. It assumes you're already very comfortable with opposition (distant opposition too) and outflanking and have worked through some basic pawn endings so you know the usual winning mechanisms.
These two examples might be better, and it gives a good lesson for practical games. The lesson is that you want to get on the same rank as the enemy pawn (first example) but opposition will only let you break through 1 rank (2nd example) and that will help you in real king and pawn endgames because you'll know to rush your king forward, and be careful to keep weak pawns as far back as possible.
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Before tackling that Dvoretskky position the examples above should be so easy they're boring to you.
Really quickly let's try the same logic with the pawns he shows, and you'll see the placement he chose for the pawns was very deliberate.
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So yes, the book might be pretty advanced, but asking these questions and looking at endgames like this is really giving you a lot of good stuff, even if you don't fully understand it yet. Definitely when you come back to these things after getting a little better they will be easier.