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I suggest you take your son to the ER. The cuts may be very close to the eye and it may require suturing. That's probably why they have sent you to the ER. The first aid
measures you have taken is perfectly alright. He may also require a tetanus
shot and some antibiotics. And even if they don't tell anything more, you can be assured that nothing more to do. You would not want to miss anything. For the swelling you need to put some warm compresses and some analgesics will help.
Hope that helped solve your query
God bless you