First time in a Digimon series there's characters using nicknames for eachother, which illustrates a special dynamics of a relationship. It shows affection from Ryo's side and annoyance from Rika's. However, this annoyance is different from what she often feels towards Kazu (Hirokazu) or Kenta, who make her mad a lot, since she stays very much indifferent to them, while Ryo just keeps bugging her. That's why Ryo's not the only one who has a nickname for Rika (in dubs), she has one for him as well. Thanks to Onkei's translations you can read about it here.
Rika first describes Ryo as the "refreshing ray of light" in episode 28 of the Japanese Digimon Tamers. It's when they first meet and he saves their lives, fixes Hagurimon and the clock - he knows what he's doing there. His confident presence annoys Rika and she mutters to herself:
Rika then uses Sparking teeth as an actual nickname for Ryo in the Digimon Tamers Drama CD: Original Story, Message in the Packet. She does so twice:
The story behind this Drama CD is happening a year after the Digimon disappeared to the DigiWorld. Henry (Jenrya) thinks he found a way to record and send a private message to their Digimon and invites the Tamers to do it. Everyone but Rika, who thinks it's useless, is willing to do it but she then changes her mind and comes to Henry's to record her message.
1) Takato hints that she only changed her mind after she met Ryo and she denies it:
2) The second time she calls him that is in her message to Renamon, who must know this is Rika's nickname for Ryo:
The next two nicknames (if we can really call them that, although they're super popular in Ryuki fanfiction) were used by Ryo in the dub series. He calls Rika a wild cat in episode 29 when he asks Takato to say hello to her:
The second "nickname" Ryo uses for Rika is in episode 51 after she agrees to give Justimon all Sakuyamon's power, which could really hurt both Ryo and Justimon.
People writing the dub scenario were without a doubt very fond of this relationship and made this scene somehow very intimate. Not only does Ryo affectionately calls her pumpkin, she too expresses her worries about him getting hurt by (gently) insulting him, staying in character that way.
There's also this one time in German version of episode 40 where Rika calls Ryo "pretty boy":
Rika and Ryo are not only great at playing the Digimon card game but also great at using the cards in an actual battle, but their nicknames - Digimon Queen & Digimon King/Legendary Tamer - are usually referred to their card game skills.
In the earlier episodes there's random kids that recognize Rika on the street as the Digimon Queen, then there's the time she first meets Renamon when a few Digimon appear in her room, asking her to be their partner - one of them calling her the Digimon Queen. Kazu calls her that in episode 28 and she calls herself that in episode 50:
German/Japanese version:
Henry says something about saving the world and if others had imagined it would end up like this.
Rika: I'm a Digimon queen when I play cards ... But here, in the reality ...
Sakuyamon: Rika, it's ok.
Ryo: You ARE the Digimon queen, Rika. But don't forget, I'm the Digimon king!
Rika: Aaah, shut up!
American dub:
Rika: How could I? I was too busy calling myself the Digimon queen.
Sakuyamon: You believed it, of course.
Ryo: Why not, you are the queen. Second only to the Digimon king!
Rika: Give me a break!
This is the only time someone refers to Ryo as the Digimon King (and it's he himself), since he's more known as the Legendary Tamer. He does so mainly to tease Rika. You know, teasing is a sign of affection.