What's the max resolution for mini DisplayPort connected to a 2013 MacBook Air?

You can get full resolution with mini DisplayPort to DisplayPort or mini DisplayPort to Dual-link DVI. Dual-link DVI supports up to 2560x1600 pixels resolution. DisplayPort supports that resolution or even higher.

You will not get full resolution with mini DisplayPort to DVI. DVI is limited to 1920x1200 or something similar. You usually won't get full resolution with mini DisplayPort to VGA.

I do not know whether you get full resolution with mini DisplayPort to HDMI. I think it depends upon your graphics card.

Bottom line. I would recommend using mini DisplayPort to DisplayPort.

You should connect the monitor through DisplayPort. You will need a Mini DisplayPort to DisplayPort cable for this.

You will be able to get the full resolution of the display this way.

I have a MacBook Pro 15" with Retina Display and a Seiki 4K monitor. The monitor works great at 2160x3840 with HDMI plugged into the MacBook Pro's HDMI port. I bought an Accell Mini DisplayPort to HDMI adapter (Accell says it works @4K and is compliant with HDMI 1.4 and DisplayPort 1.2 high bit rate). I can only get 1080x1920.

You need a Dual DVI adapter to take advantage of the full resolution of your monitor.

With a Dual DVI adapter you'll be able to use "...up to 2560 by 1600 pixels on an external display, both at millions of colours"