After Ken's self realization and the end of the Digimon Emperor's control over the Digimon World, while they are cleaning up Primary Village, Veemon notices that he can't digivolve. Kari and T.K. decide to run off and try themselves. However, Gatomon is perfectly aware that she cannot digivolve, due to the lack of Kari's crest power, and immediately quotes "No go for Gato," meaning "Don't even bother." There must have been a second pretense to this, as if there wasn't T.K. would have gone himself. This was probably an excuse to just be together (for Patamon and Gatomon too!)
Later, in the episode where Kari and Yolei realize that they are DNA partners, T.K. expresses true concern over Kari at the end while Davis is huffing off, angry (well?)
Episode 37 of Adventure 02 shows that they have matching Crests - namely, matching personality traits.
Oh yeah, Davis loves Kari. But Daikari doesn't work because she doesn't love him back. Seriously. She even says she hates him in one original Japanese episode.
Some minor Takari things seen throughout the series:
T.K. and Kari are the only ones who smile when they say "Digi-Armor Energize!" ("Digimental Up!") When they go in transport between the Digital and Real Worlds, they always go in opposite directions, while Davis goes in the front. As for their Japanese names, they are both spelled Hikari and Takeru phonetically in English, and both are spelled consonant-vowel-k-vowel-r-vowel.