The proud and divisive Corinthian church was on the brink of splintering apart.
How does the Apostle Paul appeal for its unity?
First, he calls on the people at Corinth to focus on their common destiny (1:1-9). God has chosen them for Himself, given them His grace, and will keep them strong, so that in the end, "you will be blameless on the day of our Lord Jesus Christ" (1:8). God's own faithfulness assures all this (1:9).
Second, he urges Corinth to focus on how God brought them into this destiny-- Christ and his cross (1:18-2:6). It was not their superior insight or wise words that put them on track to glory. No, it is the atoning work of Jesus Christ that did it all. At the cross, everyone is on equal ground; everyone is needy. To the perishing, this message of the cross is foolishness. But to those who believe, the cross is God's wisdom.
Third, he calls on the Corinthians to focus on how the message of the cross was brought to them (2:6-3:23). God's ideas are shared by God's Holy Spirit. The message of the cross originated with God; it comes from Him, not the world. The cross-- where a humiliated and beaten man becomes the world's only hope for salvation-- is entirely God's idea-- God's wisdom. Such wisdom runs entirely counter to the world's notion that picking up yourself by the bootstraps makes you commendable to God or self-realization brings about escape/redemption. Those who proclaim a message void of the Holy Spirit will find out the insignificance of their words at the judgment, while those who proclaim the Spirit's message will see that it lasts.
Finally, Paul tells Corinth to put the role of speakers/preachers into proper perspective (4:1-21). Christian leaders are not in competition with one another. The goal isn't to win over a clique with flowery language and demonstrate how superior you are in wisdom. Rather, preachers are called to be servants of the congregation, under God. They are not be idolized, but they should be given due honor and respect. If you think a leader isn't doing a good job, don't worry. All leaders will face God's scrutiny.
In summary, Paul appeals for unity among the divisive Corinthians by urging them to focus on their common destiny in Jesus Christ, made possible by this foolish event called the cross, which is God's wisdom conveyed by the Holy Spirit.
I think Paul's point is this: The center of Christianity is Christ and his cross. The main thing then is to keep the main thing the main thing.