It's obvious why the antagonizing/instigating, attention-seeking Cartman rips on Kyle all the time: Kyle gives him the best reactions and the most attention.
Kyle is short-tempered, and Cartman takes advantage of this every chance he gets. In "Red-Hot Catholic Love," he kept rubbing the fact that he won a bet in Kyle's face so that Kyle would continuously blow up. As soon as Kyle said, "Get up, fat boy! I'm gonna kick your ass!" Cartman stopped rubbing the money he'd won all over himself, smiled, and gratefully stood up to face Kyle's wrath.
Later, Kyle must've realized that Cartman feeds off his anger—since it showed that Kyle actually cared about the situation (about the bet; about Cartman's teasing and taunting)—because he purposefully admits he was wrong; taking away Cartman's victory. When Cartman realizes that Kyle won't give him the reaction he wants, he blows up himself, and then storms off. This time, Kyle's the one to smile, because he finally found a way to beat Cartman.
In "Mr. Garrison's Fancy New Vagina," Cartman's ripping on Kyle for not making the basketball team. When Kyle doesn't get worked up or start arguing, and his only response is to say he deserves to be ripped on, Cartman walks away disappointed; sarcastically saying, "Jesus, that was fun..." But it's not fun for Cartman if there's no response; if he can't rile up the person he's trying to get a rise out of.
(Also, Stan and Kenny were trying to cheer Kyle up in the previous scene, but they failed; and though it's likely Cartman was only trying to take advantage of a situation, he could've thought pissing Kyle off would be a good way of distracting him. That's just me thinkin' thoughts, though.)
I think arguing and exchanging insults are just how Kyle and Cartman communicate half the time, too. It's their way of making conversation; not just when they're legitimately feuding, but even getting along better (insults of endearment, if you will).
Kyle's absence in "Smug Alert," as Stan predicted, leaves Cartman feeling 'empty and hollow.' Cartman tries replacing Kyle with Butters, but Butters doesn't respond the right way when Cartman rips on him. Butters is just happy to go along with Cartman; never getting angry when he's insulted. The most he'll do is apologize. That should cause a big ego trip for the athoritah-abusive Cartman—and it probably would in any other circumstance—but Butters already has his place in Cartman's life as Cartman's lackey. Cartman realizes he needs Kyle in his life, and reappoints Butters to his old position so that he may help Cartman rescue the Broflovski's from the San Francisco smug storm. At the end of the episode, Cartman insults Kyle (probably to keep up appearances, partly, since he kept it a secret that he saved Kyle's family); and after Kyle blows up and walks away, Cartman sighs of relief and says, "That's better."
Kyle's also generally the one to try and thwart Cartman's plans; either because of moral objection or their long-standing rivalry. He'll do anything to stop Cartman, and Cartman seems to encourage this (another sense in which Cartman instigates Kyle).
In "Le Petit Tourette," when Kyle tells Cartman he won't get away with what he's doing, Cartman says, "Then the game is on, Kyle." Kyle responds by telling Cartman it's not a game, but then gets all dressed up in black and gives himself and Thomas (the boy with tourettes) code names. From this, we can surmise that they view each other as a challenge and that both enjoy competing against each other.
This episode shows that they have a strange hero-villain symbiosis. Cartman wants Kyle to try to stop him, and Kyle actually enjoys trying to stop him. Where would they be without each other other? (*Cartman looks at Butters...* "Goddamn it...")
So, not only is Kyle a challenge for Cartman, but—because Kyle initiates or responds to those challenges (as well as Cartman's insults/etc)—he pays Cartman the most attention out of all the characters, and always has the best reactions when Cartman feels like being an antagonizing asshole.
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Kyman analyzation book
RandomBasically it's pack full of kyman facts and more, Good for if your trying to learn on why people ship it or you just want more reasons to ship it ❤️💚 Enjoy...