Nature of the Church in 1st Corinthians In 1 Corinthians 1:10-12 - (“10   I appeal to you, brothers, by the name of our Lord Jesus Christ, that all of you agree, and that there be no divisions among you, but that you be united in the same mind and the same judgment. 11 For it has been reported to me by Chloe’s people that there is quarreling among you, my brothers. 12 What I mean is that each one of you says, “I follow Paul,” or “I follow Apollos,” or “I follow Cephas,” or “I follow Christ.””)

Paul begins talking about the divisions within the church at Corinth.  Were these divisions each a "house church" or were they divisions within a single church that met together?  

• Follow up question 1 - Reading 1 Corinthians 11:17-18 (instructions on the Lord's Supper) it seems that it was one church that gathered together and not house churches. If the church of Corinth was a church that met in a single location - where did they meet?  Would a home be large enough? 

• Follow up question 2 - If the church in Corinth is a collection of house churches, then the divisions clearly represent a fragmentation of the early church.  How was this fragmentation halted?  Was it to move towards a single meeting location - as evidenced by early church structures that begin to appear around 200/300 AD?(https://archive.archaeology.org/0709/abstracts/churches.html) 
https://www.christianitytoday.com/history/2008/november/why-and-when-did-christians-start-constructing-special.html
It really doesn't matter where they were meeting.  Paul addresses the letter to the church.


  Unto the church of God which is at Corinth, to them that are sanctified in Christ Jesus, called to be saints, with all that in every place call upon the name of Jesus Christ our Lord, both theirs and ours:
  1 Cor. 1:2


He could just as well be speaking directly to us as them.

To your question though, at that time they very likely would have been meeting in one house, but given the nature of the contentions (I am of Apollos, I am of Paul, I am of Christ, etc; IOW, my lineage is better than yours, naynaynay), it is not impossible that they were already meetings in different houses.

Perhaps they were meeting in the woman Chloe's house.  There is lots of speculation about who she was, but obviously she had the Apostle Paul's ear.  He trusted, he listened to her.  She was important.  Like with Christ's ministry, it was supported by very important, well-to-do women.


  And Joanna the wife of Chuza Herod's steward, and Susanna, and many others, which ministered unto him of their substance.
  Luke 8:3


If, however, they were meeting in different houses, which is very doubtful, how was the division stopped?  Clearly it wasn't.  See the letter of Clement to the Corinthians.  See also the history of Christianity.

Point is that distance, time, heritage, location is not as important as what is believed.


  Now I beseech you, brethren, by the name of our Lord Jesus Christ, that ye all speak the same thing, and that there be no divisions among you; but that ye be perfectly joined together in the same mind and in the same judgment.  1 Cor. 1:10