@Rossana43, your host is not being honest when he says "AirBnB made a mistake" if the price was wrong.
AirBnB does not set the pricing; the host sets the pricing. He is either trying to correct his own error, or to just trying get more money. (Neither of which you have to agree to.)
You say that you declined the price change, but you do not say if you cancelled your reservation.
If you have not cancelled your reservation, has the host cancelled the reservation?
If one of you has cancelled, then yes, AirBnB is going to refund you some money (how much depends on his cancellation policy and which of you cancelled.)
If neither of you has cancelled the reservation, then the host is incorrect when he says AirBnB will refund the money. If neither of you has cancelled, then the reservation is still good at the original price.
At this point, it is very important to state: you, the guest, should NOT cancel. If the host wants to cancel this reservation, then the host has to cancel the reservation, not you. The host may try to trick / convince you to cancel, but you should ignore him in that regard.
If your reservation is still good, then you are supposed to be able to travel and stay in the listing as planned at the original price. The host may grumble and about it, but he has no power to change the price without your agreement.
The host can cancel, but then he will not be allowed to rent out the listing via AirBnB on those dates, and will have other significant penalties applied. (That's why he does not want to do the cancellation.)
You can contact AirBnB support and explain the situation to them. They are the ones who can offer you some help should the host cancel on you last minute. If you can get AirBnB support aware of the issues now, it could help should the host cancel.
But if the host does not cancel, you can stay for your reserved days at the original price.