So I've seen people attaching underneath and living on the 3rd line when faced with the beginnings of 4th line territory (and have had it done to me on DGS). I did a search but could not find much information on this kind of invasion or on dealing with 4th line frameworks in general. I've read that shoulder hits don't work that well against a stone on the 4th line, so I'm curious what options there are when dealing with such a framework. Would I also be correct in thinking that someone who makes a 4th line framework (in a higher level game than ddk) is essentially inviting the invasion so as to build outside influence?

Yes, that's basically right: a position on the 4th line doesn't take a very firm grasp of the side, so often you're basically hoping the opponent will invade and live on the side, giving you more control over the center. That means that you get to a point where central influence would no longer be useful to you, you want to block off the invasion.

But that doesn't mean that 3rd line invasions can't die. Often they do (especially when you're very strong locally), so keep both options (containing and killing) in mind.

When an opponent plays a contact move on a lone stone, 95% of the time the right move is either to hane or to extend. Which is best depends a lot on circumstances, surrounding strength, reading, etc. At your level it might be a good idea to automatically hane every contact move, and see what sort of situation develops out of that. Then experiment with automatically extending in response to every contact move.

Just a quick comment that attaching under a 4th line stone is a rather unusual way to invade. The normal sort of shape is a knight's move on the 3rd line from a 4th line stone, such as a or b below in a continuation from san-ren-sei (c is another invasion point, but being a corner is rather different and easier to make life). Of course there are also 4th line invasions (one point to the left of a/b), the general principle is these are to either to be able to run away faster, or split your opponent and get attacks going. Third line invasion on the other hand lets your opponent connect on top if he wishes (so you aren't attacking), but aims more to live on the side.

Click Here To Show Diagram Code

[go]$$c San-ren-sei invasion points
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . O . . c . . |
$$ | . . . O . . . . . , O . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . a . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X , . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . b . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . O . . O . . X . . |
$$ | . . . . . . . . . . . . . . . O . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Finding invasion points is of course not the end of it. There's also picking the right one (for example a is better than b due to the open corner above, see my lecture on this shape at viewtopic.php?p=86510#p86510 for explanation). And then finding the right time to invade. And of course reading likely continuations.

Invading frameworks based on the 4th line is a huge topic. I could talk about it for 100 hours, and my knowledge is a mere fraction of the total. Off the top of my head, books that cover this sort of thing include Keshi and Uchikomi: Reduction and Invasion in Go, Get Strong at Invading, any game commentary book such as Tournament Go 1992 will surely feature some invasions of 4th line frameworks (maybe not so many in Invincible, games of Shusaku as 4th line not played so much back then), and there are some bits in Yilun Yang's Fundamental Principles of Go.

As you say, shoulder hits are usually against 3rd line stones not 4th line ones (as just blocking builds an extra line of territory), though of course there will be rare situations where shoulder-hitting high stones is a good idea, for example as a leaning attack (I recall seeing a game where Takemiya shoulder hit a 4-4 at 5-5 as part of an attack). The big thing you didn't mention about build a 4th line framework is ATTACKING. Sometimes you will seal in the invader and let it live in exchange for outside influence. Other times you will stop it getting 2 eyes on the side and let it run away, but as you attack it you will gain territory here, strength there, endgame here, the option to invade your opponent's formerly secure territory over there etc. A good attack can have a profound influence on the entire game.

Thank you both very much for your comments.