multiple pairs of wireless devices operating in the same space

I am working on a project that involves two transmitters sending a signal to a receiver that is no more than 7 meters away from the transmitters. The signal that needs to be transmitted only has to indicate if a button was pushed and which of the two transmitters it is coming from. However, I will need multiple sets of these (about 6 sets) to be operating in the same room (about the size of a high school gymnasium) without affecting each other. What would be the best way to send the signals? i.e. radio, bluetooth, etc. I was leaning towards radio, however most radio transmitters/receivers I can find for the arduino only operate on 315 or 433 MHz, which seems like it would limit to only two distinguishable signals. I have moderate experience with arduinos but no experience with wireless communication. Is there any way to have this many devices operating in the same room without interfering with each other? Are there any components or similar products anyone knows of? I have tried looking into this and have only found unrelated projects/products. Any help would be greatly appreciated.

I think you have several options.

I think the inexpensive nRF24L01+ transceivers (I call them "Nordic modules") would likely work.

Here's a panic button project I did several years ago. I don't think you should have a problem with six sets of transceivers.

How much data needs to be sent? How important is the timing?

These Nordic modules are actually fast enough to be used with as camera flash transmitter (with some clever programming) so there isn't particularly a time delay issue my main concern is if this large room is full of people. I can receive signals between two Nordic modules anywhere in our house except where there's a large appliance in between the transceivers. But sometimes the transceivers miss signals when they're in the same room if a couple people are standing in between the transceivers.

If timing isn't important you can increase the chance of a message getting from one device to another by using the other transceivers in the room as repeaters. This complicates the programming but greatly increases the odds a message gets through.

Have each transmitters send a unique ID using morse code. The receiver needs to decode the message to figure out which transmitter is active.

A lot of modules allow you to address the target transceiver. The ID process can stay internal to the transceiver itself.

I was leaning towards radio, however most radio transmitters/receivers I can find for the arduino only operate on 315 or 433 MHz, which seems like it would limit to only two distinguishable signals.

Only if the transmitters are on continuously.
The dirt cheap 315 or 433 MHz combos work very well with the VirtualWire library, using very simple programs.

I suggest to send a two-byte message with transmitter ID and a button indicator, which would take a few milliseconds. Using that approach, 50 transmitters could be operating at random intervals and almost all of the messages would arrive successfully, at any single receiver.

Or, if you just want to know when a button is pushed, the transmitter sends a one byte ID twice, with a short delay (unique for each transmitter) between transmissions. That way if two people push buttons at the same moment, the second transmissions from each transmitter are guaranteed not to overlap. VirtualWire rejects overlapping messages.