In the first few centuries of Christianity we have recorded instances of the Bishop of Rome exerting influence beyond his See and the comments of other Christians about the Bishop of Rome. The question is, how do we interpret those events? In the following, I try to fairly give every possible interpretation from both the Orthodox and Catholic side to assist in putting the events in context.

For a compilation on the early Papacy, see The Early Church Fathers on the Papacy: An Orthodox Quote Mine.

YEAR RELEVANT EVENT OR QUOTATION POSSIBLE INFERENCES
70s-80s THE CORINTHIAN SCHISM

Clement of Rome writes, “If anyone disobeys the things which have been said by him [Jesus] through us, let them know that they will involve themselves in no small danger” (1 Clem 59).

 -Clement was writing on a subject already clearly written about by Paul in 1 and 2 Corinthians. He was simply reminding the Corinthians of the general principle that Christ laid down: if they do not repent, with humility, of their schism damnation is probable.

-Clement had jurisdiction over Corinth.

-Clement had no jurisdiction, but as a confidante of Paul’s and representative of a notable church his letter carried weight.
100s-110s IGNATIUS’ LETTER

“You [Rome] have never envied any one; you have taught others” (Ignatius to Rom, 3).

 -Rome was a respectable church and these are laudatory words, given in light of the fact that historically men like Paul, Peter, and Clement taught there.

-Rome had a teaching authority over other churches.
170s-190s COMBATING GNOSTIC VIEWS OF AUTHORITY

“Peter and Paul, that church which has the tradition and the faith which comes down to us after having been announced to men by the apostles. With that church, because of its superior origin, all the churches must agree, that is, all the faithful in the whole world, and it is in her that the faithful everywhere have maintained the apostolic tradition” (Irenaeus, AH 3.3.2).

 –Because Rome’s succession of Bishops have preserved, historically, the teachings of both Peter and Paul, making it incumbent upon Christians everywhere to maintain the same traditions/interpretations.

-The “superior origin” is a reference to Peter and Paul’s teaching at the church, not a reference to a supernatural teaching office bestowed by specifically by Peter to a specific Roman Bishop.

-The “superior origin” is a reference to an infallible teaching and administrative office bestowed by Peter to all his successors, hence agreeing with Rome is absolutely imperative.
190s PASSOVER CONTROVERSY

-Pope Victor I threatened to excommunicate the Bishop of Ephesus because he celebrated the Passover. He was scolded by Irenaeus and avoided breaking the peace.

-Pope Victor I excommunicated Theodotus the Tanner/Shoemaker for Christological heresy

 -The Bishop of Rome asserted the power to excommunicate believers anywhere in the Christian world, and this was rejected.

-The Bishop of Rome asserted the power to excommunicate believers anywhere in the Christian world, and this power was recognized as legitimate but it was the wrong situation to exercise it.

-The Bishop of Rome only had power to break communion in his local church, evidenced by both the excommunication of a local Greek named Theodotus and an earlier episode where Anicetus relented and gave communion to Polycarp despite disputing over the same issue (“Anicetus conceded to Polycarp in the Church the celebration of the Eucharist, by way of showing him respect,” Irenaeus Fragment V).
199-217 MONARCHIANIST CONTROVERSY

Caius, a priest from Rome, wrote: “[T]he apostles themselves, both received and taught those things which these men [followers of Theodotus the Tanner] now maintain; and that the truth of Gospel preaching was preserved until the times of Victor, who was the thirteenth bishop in Rome from Peter, and that from his successor Zephyrinus the truth was falsified. And perhaps what they allege might be credible, did not the Holy Scriptures, in the first place, contradict them [in their doctrines]” (Fragments).

 -It is a credible notion that a Bishop of Rome, with valid succession, can lapse into heresy in the mind of Caius.

-We are taking Caius words too literally–he is merely making a rhetorical point similar to how in modern conversation one may say, “even conceding you this, you would still be wrong because of that.”
250s NOVATIONIST CONTROVERSY

Cyprian wrote the following when confronting the Novationist controversy after quoting John 20:21 and Matt 16:18: [Christ said this so that] “He might set forth unity, He arranged by His authority the origin of that unity, as beginning from one. Assuredly the rest of the apostles were also the same as was Peter, endowed with a like partnership both of honour and power; but the beginning proceeds from unity…And this unity we ought firmly to hold and assert, especially those of us that are bishops who preside in the Church, that we may also prove the episcopate itself to be one and undivided (Unity of the Catholic Church, 4-5).”

 -The authority given to Peter has been, in all ways (“both honor and power”), equally given equally to all the Apostles and succeeding Bishops.

-The honor and power of Bishops is only effectual if they maintain unity and avoid schism. The Novationist question was over whether unity should be maintained with the Bishop of Rome. Cyprian’s answer was categorically “Yes”, as only Bishops in unity exercised the office of the keys.

-Peter is preeminent because unity has its source in him. Bishops are equal in their capacity as Bishops, but unequal in their authority over other Bishops.
250s REBAPTISM CONTROVERSY

Bishop Stephen of Rome permits the lapsed to re-enter the church via Chrism. This was uncontroversial, as most of the Church did this. Those who did not, according to Cyprian, were in their right to do so: “[E]very bishop disposes and directs his own acts, and will have to give an account of his purposes to the Lord” (Letter 51.21).

The matter of dispute was both over the baptism of schismatic heretics (the Novationists, who broke with Apostolic Succession by declaring the orders were ruined by lapsed clergy and so installed Novation as Pope and split existing congregations by re-baptizing Catholics into their own church) and the baptism of Christological heretics such as Marcionites. Pope Stephen also allowed lapsed clergy to serve without penance; see Dionysus of Alexandria, Letter III; Cyprian, Letter 73.2, 7, Cyprian, Letter 71.1-2; Cyprian [Firmilian], Letter 74.5).
Pope Stephen’s both condemned the Novationists (for schism) but also “condemned” in “his letters” Cyprian and a council of African bishops which opposed the taking-in of Marcionites, presumably acting in his capacity as Bishop (see Cyprian, On the Baptism of Heretics, Preface). According to Cyprian, Pope Stephen cited the tradition of the heretics to bolster his more lax view, writing: “Since those who are specially heretics do not baptize those who come to them from one another, but only receive them to communion” (quoted by Cyprian, Letter 73.4). Firmilian concurs citing the existence of the same passage, “Stephen said in his letter that heretics themselves also are of one mind in respect of baptism” (Letter 74.7) and the (same?) letter called Cyprian “a false Christ and a false apostle, and a deceitful worker” (74.26).

Stephen allegedly almost caused a schism over this issue: “For what is more lowly or meek than to have disagreed with so many bishops throughout the whole world, breaking peace with each one of them in various kinds of discord at one time with the eastern churches, as we are sure you know; at another time with you who are in the south, from whom he received bishops as messengers sufficiently patiently and meekly not to receive them even to the speech of an ordinary conference” (Letter 74.25).

Cyprian’s council in Carthage appears to reference the same behavior accounted to Stephen that was referenced in Firmilian’s letter: “For neither does any of us set himself up as a bishop of bishops, nor by tyrannical terror does any compel his colleague to the necessity of obedience; since every bishop, according to the allowance of his liberty and power, has his own proper right of judgment, and can no more be judged by another than he himself can judge another” (On the Baptism of Heretics). He also accused Bishop Stephen of “human error” (Letter 73.10).

Firmilian wrote that “they who are at Rome…vainly pretend the authority of the apostles” (Cyprian, Letter 74.6)

 -Stephen was in obvious error and his stance of not baptizing Christological heretics was rejected by the Council of Nicea (Canon 19), Council of Constantinople (Canon 7) and modern Roman Catholic practice (Mormons are not chrismated). The Christian world of Stephen’s time did not recognize his infallibility.

-No Bishop was viewed as having supremacy over another Bishop, which explains Cyprian’s reluctance to reject Bishops who rebaptized Novatians, a practice he personally opposed (Letter 73.12).

-Consensus can overrule a Bishop of Rome.

-The whole Christian world was wrong and misrepresented Stephen as using his capacity as Bishop in accepting Marcionites without baptism.