So I am remodeling and adding an island with a sink. I need some help in figuring out how I can plumb this correctly. I think I understand how to do the bow vent or loop vent technique but in my area there is another way that is allowed I feel would be easier but I'm not quite sure if I can get it to work with what I have. The method I would like to use is a 3" drain. I'm assuming this works by over-sizing the drain so it sort of vents itself?

I will try to post a few pictures but let me explain best I can. The kitchen in on the main level and the stack I want to use is a 2" that runs down into the basement and the into the concrete floor. I'm assuming that I will have to change the stack so that the drain running down into the basement will have to be 3" all the way down.

I took a picture of where the drain runs into the concrete and what I think is a clean-out that measures 4" round. could I just tie in a new 3" drain into the old 4" clean-out?

if your code allows what you have drawn then go for it..
as for using a cleanout for the tie-in point, that is fine as long as you replace the screw in plug with tee or 'y' so you still have full size cleanout capabilities.

The cleanout opening is probably a 3 1/2" plug so you would have to find a 3 1/2" x 3" bushing.

The diagram shown was sent to me by the local plumbing authority as an alternative to what is allowed for island sink drain. My island sink won’t line up perfectly straight with the stack but I will only have to have one bend in it to reach the stack. I will make sure to add a clean-out at the bottom where I plan on connecting to the old clean-out. I will try to draw up a diagram just to clarify.

I’m shocked this unvented drop down then turning horizontal under the floor is allowed under any code.

I guess they just think that there is enough volume of air in a 3 inch pipe to offset any syphoning problem that a 1 1/2 inch trap opening could develop ?

Seems like it would be easier to go 1-1/2" and stick an AAV under the counter

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Not a plumber by trade but a fierce DIYer

Turns out I may have a pretty straight shot to the 3” stack under the floor so i may not have to have any turns at all. I have hard a hard time finding any information about venting an island this way but I did find a thing or two about using an oversized drain to essentially vent itself. I’m no plumber and definitely not very versed in the DWV aspect of plumbing. I know many people, especially plumbers talk bad about aav’s but to me I think they would be nice to use in situations like this. I know my area uses 2009 uniform plumbing code but apparently this 3” vent for island is also locally allowed.

I like the 3 inch pipe better than the AAV.
just a personal preference.

will a disposer put enough pressure to close off the air space in that horizontal section of pipe?

the disposer elbow is 1 1/2 inch OD so the ID is probably closer to 1 3/8 inch. I can't imagine that filling up a 3 inch pipe.

Well, I am planning on having a disposer and also tying the dishwasher drain into the disposer. I would assume that would be ok. Also, in the diagram it shows a 3x3x2 sanitary tapped tee. Is this just the 2” part of the tee is threaded?

in plumbing dialogue, a tapped fitting is a female threaded fitting.
you want a glue in fitting or sometimes called 'socket weld'.
it is hard to tell from your diagram but for a double bowl sink you can use a 3 x 3 x 2 x 2 double sanitary tee with separate traps for each bowl.