In Saint Luke's Gospel 22:24ff. we read:
And there was rivalry between them over the question, which of them was to be accounted the greatest. 25 But he told them, The kings of the Gentiles lord it over them, and those who bear rule over them win the name of benefactors. 26 With you it is not to be so; no difference is to be made, among you, between the greatest and the youngest of all, between him who commands and him who serves. 27 Tell me, which is greater, the man who sits at table, or the man who serves him? Surely the man who sits at table; yet I am here among you as your servant. You are the men who have kept to my side in my hours of trial: 29 and, as my Father has allotted a kingdom to me, so I allot to you 30 a place to eat and drink at my table in my kingdom; you shall sit on twelve thrones, judging the twelve tribes of Israel. 31 And the Lord said, Simon, Simon, behold, Satan has claimed power over you all, so that he can sift you like wheat: 32 but I have prayed for thee, that thy faith may not fail; when, after a while, thou hast come back to me, it is for thee to be the support of thy brethren.
I've seen this part used to challenge the Papacy. The question I ask is: Why did the disciples dispute who was the greatest, if Peter had been made Christ's Vicar?
That's the general question. However, second one exists too.
Considering verses 29-32, where Jesus seems to be pointing out that Peter is indeed special, the question might be framed differently: About what kind of greatness are the disciples disputing, before Our Lord tells Peter what his job is?