which ryzen 3 for simply gaming with a gtx 1060

Solution
Uh, you act like an R3 is plenty of CPU processing power but it's not.

It varies by the game but unless your budget is severely limited I'd go for the R5-2600 (6-core/12-thread) which is only about $165USD.

I wouldn't get anything less than an R5-1400 but it's pointless as the cost is almost that of the R5-2600, but frankly an R3-1200 (4C/4T and lower frequency) is about $95 and an R5-2600 (6C/12T higher frequency) is about $165 you're paying about $70 extra for a far better CPU than the R3-1200.

But if budget is tight sure go for the R3-1200 and upgrade later but some games will have more stutter or lower FPS.

Other: With RYZEN you should get:
a) 4x DIMM slots, and

b) 2x4GB 3000MHz CL16 (roughly)

Ryzen needs fairly high bandwidth...
Uh, you act like an R3 is plenty of CPU processing power but it's not.

It varies by the game but unless your budget is severely limited I'd go for the R5-2600 (6-core/12-thread) which is only about $165USD.

I wouldn't get anything less than an R5-1400 but it's pointless as the cost is almost that of the R5-2600, but frankly an R3-1200 (4C/4T and lower frequency) is about $95 and an R5-2600 (6C/12T higher frequency) is about $165 you're paying about $70 extra for a far better CPU than the R3-1200.

But if budget is tight sure go for the R3-1200 and upgrade later but some games will have more stutter or lower FPS.

Other: With RYZEN you should get:
a) 4x DIMM slots, and

b) 2x4GB 3000MHz CL16 (roughly)

Ryzen needs fairly high bandwidth to DDR4 memory so I'd not go too cheap on the DDR4 memory. A very poor choice would be 1x2400MHz.

Anyway, it would be best to give the total HARDWARE budget to play around with a PCPARTPICKER build.

I'd also try to start with a GTX1050Ti if possible as that has 4GB of video memory. If you can find the cash it's a huge difference to have R3-1200 + GTX1050 vs an R5-2600 + GTX1050Ti, not only will that 6-core CPU last a long time but you can also wait to upgrade the video card a bit longer too... basically if you got the R5-2600 I'd only get another identical 2x4GB DDR4 kit later then upgrade the graphics card in the future to a GTX1170 or whatever comes out.
 
Solution
I would agree with most of what photonboy says. surely get two sticks of ram as a single stick hurts performance and getting something that can OC/XMP 2933+ helps cpu performance and a 1050ti at 4gb is the better choice than a 2gb 1050, just a better overall initial experience.

Now i feel a ryzen 3 would suffice, but not a 1200 as the clock speed is lower and it is fairly easy to OC but some people do not want to, so a 1300x or 2200g would be good. I have a system with a 2200g that OC to 4.1ghz with a 780ti which is roughly similar to a 1060 and get similar performance with the 780ti as my 5820k does in a few games that I tested.

Now I do still agree with photonboy that a ryzen 5 would be the better choice in the long run, and for the cost of about $150-170 you can get a 1600/2600 which is not that much more than a 1300x or 2200g, but I do understand being budget limited. Just to add my two cent.

Edit: my buddy runs a 3.8ghz 1400 at 2933 ram and a 6gb 1060 and it works great for him, so if possible and the 1600/2600 is out of reach the 2400g/1400 is also a good option.
 
Depends on the games you want to play. Most games will be limited by your videocard at 1080p as long as you have a modern 4 core/4 thread CPU like a Ryzen 3.

There are exceptions. For instance, Battlefield 1 multiplayer, and I assume all future Battlefield games, will need more than that if you want to avoid choppy framerate in large multiplayer matches.

My personal preference is to invest a bit more in the CPU at the start, then upgrade the videocard over time as needed. This is because between the two, CPU and videocard, the videocard tends to be the one that gets outdated first. If you bought a Ryzen 2600 today and a 1060, the 1060 is the one you'll need to replace first, I can almost guarantee that. Another reason is videocards tend to change the most over time, so you tend to get more out of a videocard upgrade. CPUs can go years before you see any real big change. People are still gaming with Sandy Bridge and Ivy Bridge, but I bet most if not all of them have newer videocards.