It's always a bad idea to lie to the police. Even the guilty are much better to not say anything than to lie.
That being said, absent other facts which you haven't told us, there could not be a conspiracy for you telling the police you did not know the person after the fact. Accessory after the fact, perhaps, but not conspiracy. Obstruction of an investigation is also a possibility. However, the police would need to prove you knew the person. One way is the phone records. To get the records, absent some modern fancy-law exceptions which don't apply, they would need a search warrant. This would require an affidavit showing probable cause that you committed a crime (or that the records would elicit relevant evidence). The difficulty here is, at least based on some assumptions from your post, the police would have to provide some evidence you knew the person. (Perhaps another at the fight who said you were the suspect's friend.) They then would need to write up an affidavit to show why they believe the information they have gets to the level of probable cause. They then get the phone records, take the last few or all of the entries and try to identify the person(s) associated with the numbers. They then question those people or get ID photos from a driver's license data base and do a photo line up or other investigatory technique to the victim or other witnesses to identify the suspect. Whew.
They are not going to do this if it was just a fight. If the victim was seriously hurt or if the fight had other events/facts related to it (extortion, drug sales etc.) they might because of the seriousness of the crime. The bottom line is that they would probably not do what they are threatening to do. But if they do feel it is worth the effort, you are going to be charged with as much as they can charge you with.