Monitors that are compatible

The short answer is almost any, with an adapter to connect your Mac's port to the monitor (e.g. you might need a USB-C to HDMI cable).

However, with any screen under 4k resolution the issue will be text quality.

The text will look fuzzy, unless you can use an HiDPI resolution - and that will always be less than the monitor's native resolution.

For example, on a 25" BENQ QHD monitor (2560 x 1440), I get HiDPI at 1920 x 1080, which looks good.

On a Philips 22.5" FHD monitor (1920 x 1080) I can only get HiDPI at a too low resolution (everything looks too big).

The "issue" is explained here: https://github.com/waydabber/BetterDisplay/wiki/MacOS-scaling,-HiDPI,-LoDPI-explanation

Which 13" MBA do you have? Click on the Apple menu--> About This Mac. Then let us know which model you have, including the year. (It's the only way we'd know what ports you have on your MBA).

I have a 4K LG monitor that works just fine.

One thing to do, toss the garbage cables supplied by the manufacturer and purchase certified cables for the connection you intend to use. Display manufacturers will put the cheapest cable they can get away with in the box and their reliability can be suspect.

You didn't say what size monitor you're looking for.

This is a good YouTube video showing how two different monitors, one of them 4K, the other 1440, look when using them on Macs. (Keep in mind the monitors here are 32"). Monitor Comparison Video

I have a 4k monitor and I'm getting display issues, as for text it looks fine, but I have a horrible green tint, to ally my imiages, green, pink, etc... the color profile is way off