I learned from the Internet that the safe current of 1 mm2 copper wire is about 4-5 Amps (DC).
Well there's your problem then.
4 to 5 A/mm2 is a rating, and a very conservative one at that. It makes all sorts of assumptions about the thermal environment and cooling. That sort of rating would be applicable for the copper in the windings of a small transformer (lots of heat, little cooling). You could use a 10 A rating for multiple plastic insulated wires run in a conduit.
When you get to control the thermal environment, it's only temperature that limits what current copper can carry. That means balancing the heat produced by the current with the heat lost to ambient.
What temperature is the conductor permitted to reach? Is there plastic insulation on it limiting it to maybe less than 100 C, or is it bare metal and can reach several hundred C without damaging its environment? The hotter it can get, the more heat will be lost per unit area, and the smaller conductor you can use.
Unfortunately, calculating the rate of heat loss from a conductor is a complicated affair, involving serious fluid dynamics. You should though be able to find tables online that will give you estimates for flat plate heatsinks that will give you a ballpark to work in.
Frankly, the best way for an amateur to do it is make a test conductor, heat it with a test current, and measure it in situ. Compute the resistance at RF for your test conductor taking skin depth into account, and multiply by your intended current squared to get the power you're going to dissipate. Now heat it with the same power, from RF if you have it available, otherwise DC or mains frequency (making the appropriate adjustments for lower resistance) and see how hot it gets. You could use a low power to simply estimate a Watts/C, which will be fairly linear for small temperature rises, but will be non-linear as the cooling air becomes turbulent or IR radiation becomes more significant.
