I recently bought a new aquarium, along with four tiger barbs. I have already introduced two new Rosie barbs into the tank with them.
I'm now wondering if I can introduce tetra fish into tank. And if yes, then which type of tetra should I introduce? How many of them minimum should be kept together?
No. Definitely not at the moment, but possibly later on down the road.
I assume this is the same tank you ask about here and here. This is a new tank that does not have an established biofilter yet: this means the more fish you put in now, the more unhealthy the water will get. Until you know for a fact that your biofilter is working -- your ammonia and nitrite levels stay at 0ppm and do not fluctuate -- there is no point in adding any more fish. If you do, the most likely result is that they'll die, or suffer permanent damage to their gills that greatly shortens their life.
But can you add tetras once the biofilter is working? The answer to that is 'maybe'.
There are dozens of species of tetra out there. I don't know what's available to you or what kind of fish you're looking for, so I'm not going to make a specific recommendation. But basically, you need to ask two questions to decide whether you can keep fish together: how will they interact with each other? And, can I provide environmental conditions that are healthy for both species?
In terms of behavioral compatibility, tiger barbs are active, somewhat aggressive fish. I've even heard some people say they don't believe they're suited for a community tank at all. As Seriously Fish notes:
'P.' tetrazona is notorious as an aggressive community inhabitant with a reputation for nipping the fins of tankmates though this behaviour only tends to be pronounced when insufficient numbers are purchased or space is limited... That said it is relatively boisterous and doesn’t make an ideal companion for very shy, slow-moving, or long-finned fishes such as many livebearers, cichlids, and anabantoids.
Tetras are often relatively shy, so in your research you'll want to make sure a species can handle being pushed around a bit. The flip side is that there are very few tetra species that will cause problems for the barbs. And as the SF article mentions, you'd want an active, short-finned species.
In terms of environmental compatibility, it looks like tiger barbs and most tetras prefer slightly acidic, soft water, so that's a plus. The SF article mentions the barbs come from clearwater streams and rivers with a sandy bottom and thick vegetation, with temperatures in the mid-20s (°C)/mid-70s (°F). I would rule out the tetras native to blackwater conditions, where the water is still and warm, dark with tannins, with little live vegetation, but there are still plenty of species to choose from. So that's what would help you pick out a tetra that's compatible in terms of water conditions and behavior.
But there's another aspect to environmental compatibility: once you know what the fish need, can you provide it? The biggest limiting factor here is your tank size. Tiger barbs and most tetras are highly social and are best kept in large groups, which means you would need a good amount of tank space and filtration capacity. I don't think I've seen you mention anywhere how large your tank is. The four barbs you have will get pretty large -- 3" (7.5 cm) or so. If you have a small tank, like a 10 gallon (38 liters), you probably don't have room for any more fish.
Finally, consider alternatives as well. The barbs will spend most of their time in the middle of the water column. You can avoid a lot of aggression problems by choosing tankmates that stay towards the bottom -- I think loaches are common tankmate, for example.
