Charlie we use number 8 hardware cloth here in America. 0.125'' 1/8'' which equals = 3.18 mm.
It is my understanding this is the smallest opening suggested for the purpose you seek.
Since number 7 hardware cloth is 3.629 millimeters, that will mean you will be pushing it at 4mm.
Assuming the following information is 100 percent accurate. I would like to hear from other beekeepers who actually use or have tried 4mm and hear what they say.
I found the following in the archives here at Beemaster. The following was speaking in answer to 1/4 inch hardware cloth for another purpose. Not the same question you ask, but the answer should shed some light to your question Charlie.
Phillip
very easily, they have no trouble going through it, most use it for mouse guards, you will need number 7-8 hardware cloth for bee's not to go through, some go through number 7 but 8 is to small. think I got my numbers right, sure some will correct me if I am wrong, but 1/4 inch bee's will go through with no problem.