Ross seems to have a childlike need to always be right. Such as in The One Where Heckles Dies, when he finds out that Phoebe doesn't believe in evolution. Shocked by this, he spends the rest of the episode pestering her and trying to get her to believe in evolution. He is angered when Phoebe asks him why he has this need for everyone to always agree with him. He seems to find people irrational if they won't conform to his point-of-view.
In most cases, his arrogance backfires or gets the best of him. In later seasons, due to his two failed marriages and ruined life, he starts developing breathtaking anger management issues and also shown to easily be irritated, often breaking into angry outbursts over minor things, even showing signs of sociopathic behavior. For example, in The One With Ross's Sandwich, Ross is fired from the museum after yelling at his boss for eating his sandwich. This behavior leads to him taking on a more antagonistic role in most episodes in the later seasons.
Ross' childishness was more emphasized and apparent at the beginning of his and Rachel's relationship in which he began spending (if possible) even more time at Monica's apartment and their relationship reverted back to when they were teenagers which in the same episode Monica reveals she hated him during their youth. However this isn't unique to one particular episode and they have at times shown traits of typical sibling relationships even at their adult age throughout the series; rash outbursts, competitiveness, and physical confrontations are examples of such. Perhaps the cause of his childlikeness is his parents. As he was the favored and academically gifted child who received the more credit and praise in comparison to Monica, he can't always be blamed for his sometimes enlarged ego.