This is the part of DigiRoyalty that I used to call "Evidence" to prove that Ryuki is the one true romantic relationship in Digimon Tamers. I would point out things like "Ryo is the only one who follows Rika down the chasm!!11!1" or "Rika didn't cry when Beelzemon was shooting at Takato but cried when she gave her powers to Justimon!1!!" and it's something I've been trying to avoid in this 2020 makeover of the page. :D I still really enjoy this relationship and don't understand what Rukato fuss is all about (although that tunnel scene is indeed weird) but I decided to just re-explore the relationship between Rika and Ryo and point out what I think are its main characteristics.
Rika and Ryo have a history together, they are both skilled Digimon TCG players and fought in the finals of a cardgame tournament. Ryo won the finals but disappeared to the DigiWorld afterwards. With him out of the picture, Rika won at least one tournament and was crowned the Digimon Queen. There's people like Kazu who believe she could only become number one after Ryo disappeared.
After meeting Renamon and becoming a Tamer, Rika (and Renamon) are fixated on becoming stronger and stronger, defeating and loading every Digimon they fight with. Winning is something Rika's obsessed with until she befriends Takato and Henry and realizes how much she cares about Renamon. However, she still takes pride in her cardgame skills and when she unexpectedly meets Ryo, her number one rival & one of the reasons behind her fixation to be the best, she instantly hates him for it. Everything about him pisses her off and she can't even hear his name without getting super annoyed.
He poses a threat to her inner peace, to the part of her that gave up on wanting to be the best. But then she meets him and suddenly she's set back to the situation where people doubt she's better than him. Ryo on the other hand doesn't care about their rivalry. He doesn't even remember her at first but later when he sees how strong and courageous she is, he's not intimidated but rather amazed by her.
He doesn't see Rika as a rival because he doesn't feel the need to prove himself to anyone. His skills are a result of lots of experience he's had with Digimon (in his other dimension mostly) and he uses them to help Cyberdramon with his quest. Rika on the other hand was trying to build her identity and prove herself in a world that wouldn't just accept her for what she was. Her father (who died/left them) wanted a boy while her modelling mother wanted a girl that would follow her lead and have a thing for cute dresses. Rika as a normal rebellious kid found her purpose in Digimon fighting, a domain usually reserved for boys, something that surely spured her fixation on getting stronger and stronger and continue winning. And that's something Ryo as a guy wouldn't experience.
But Ryo's not only that one guy she couldn't defeat once and makes her look second-best, he also seems annoyingly perfect. He's good looking, nice, experienced and popular. His refreshing personality that's such a contrast to hers pisses her off so much that she gives him a nickname.
However, her initial hatred softens after episode 39 when she and Renamon evolve to Sakuyamon and the need of winning and being the best is once and for all overcome by the need of protecting the loved ones. Her rivalry with Ryo isn't important anymore and Rika proves it by giving Justimon all Sakuyamon's powers and letting him be the one to defeat the enemy in episode 51.
With her insecurities gone, Rika accepts Ryo as the good stand up guy he is but can still get annoyed by his goody-goody image and (sometimes obnoxious) proneness to sudden dramatic appearences (he does love to show up unexpectedly). And she always expresses her annoyance, for which we have to love her so much (RUKI (episode 46): Yeah, I get it. There's only one person with that annoying "refreshing" presence.").
She often uses sarcasm when talking to him. She has some pretty savage comments in their initial conversations but they become nicer and are often followed by a small laughter later:
VS
Unless he really touches a nerve:
Rika accepts Ryo as one of the people she cares about, even though she can still have a problem admitting it outloud:
According to Digimon Tamers' story developer Chiaki Konaka, it was decided beforehand that Ryo, the popular protagonist of the Digimon WonderSwan games, would appear in the third season of Digimon. He's introduced in the second half of the series as an experienced Tamer, well known in Digimon TCG community as the legendary Tamer who disappeared after a Digimon tournament. Ryo's tied to Tamers' universe through Digimon TCG, while he maintains his connection to WonderSwan games through Cyberdramon's inner obsession with finding his arch-enemy. These two things define him as a character in Digimon Tamers.
I noticed that a lot of people dislike him because his character isn't established enough and he feels like a leech whose identity is shaped only by his relationship with Rika. I agree that we don't get to know Ryo enough and that he's way more interesting (and interestingly less out of space) if you know about his WonderSwan background. However, even before knowing about that I never saw him as an intruder because he never really felt like one of the team. Sure he was fighting against the D-Reaper with Rika, Takato and Henry while Kazu and Kenta were more or less pushed to the side and turned into a comic relief, but he was kind of just there, he didn't try to replace anyone and even had his own reasons to be there (not just to save the world but to continue Cyberdramon's quest). He was one of the four Tamers able to evolve to Digimon Mega level but he was absent for all great scenes the trio had bonding, he never interfered. For me it felt like he had his own (untold) story that only connected to the rest of the Tamers' story at certain points. Plus his feelings towards Rika are way more subtle in the original Japanese version than they are in the dubs, where they really made a big deal out of them. I understand why he's not a good character story-telling-wise but I still think the writers did a good job introducing him into the Tamers' universe without making the season focused on him (which would probably had to happen if he was around more, since he was the protagonist of the popular games, right?). I do wish they'd show more of his amazing skills rather than just talk about them and just showing us that one scene where he uses a super-duper-amazing card but okay ... you know what would also be pretty cool? If there was a hint or two about his existence in the earlier episodes - like for example a "missing person" poster somewhere in the background or sth like that.
Anyway, most of Ryo's relationship with the Tamers is focused on Rika (and Kazu). While he does exchange a few words with Takato and Henry, it's Rika and Kazu that stand out the most. While Kazu and Ryo's relationship is more or less a onesided admiration purposing a comic relief, Ryo's relationship towards Rika has more than one characteristic, which are: Ryo being worried about her, accusing her of being reckless and stubborn, trying to help/save her, being nice to her and then baffled because she can't stand him and says something mean, admiring her and even teasing her to show his affection for her. However, it wouldn't be right to say he does all that only to impress her.
Ryo may be a laid back guy with a refreshing personality but he's not joking around when things get serious. There's just this one scene in episode 50 where he calls himself the Digimon King only to tease Rika, which makes the serious moment before the last battle a bit easier, especially when Takato jumps in on it. But other than that, he's worried when Rika does seemingly reckless things, accuses her of that but then realizes how strong and courageous she is and isn't trying to stop her anymore (something all of the Tamers had to learn at some point).
There's this inside joke between them I completely missed until now, based on Ryo constantly accusing her of being reckless/stubborn:
It's in the cute moment they have when Rika agrees to give Justimon all Sakuyamon's powers and tears come down her cheeks when she tells him how he turned out to be the reckless one of the two (which he takes as a compliment):
It's obvious that Ryo likes her and willingly annoys her a few times to express his affection. Like when he asks Takato to say hello to her, knowing she wouldn't be too thrilled about it, or pulling her headphones down in the first Drama CD. He teases her in both Drama CDs and is even straight up flirting in the second one, Digimon Story Special ~with~:
However, he teases and is way more flirtatious in the dubs. The Ryuki fan in me loves the extra weight of their relationship there but can also understand why people, who don't care about this relationship, find Ryo annoying (and leeching onto Rika's character). More about the dubs in Comparison.
This was a short analysis of Rika and Ryo's relationship, combining everything I found out from the show itself and from the two Drama CDs following it. I just wanted to have something like this in one place so that I don't have to analyse the episodes separatedly but rather just point out what I've been describing here.
1) Rika loses the finals of the Digimon Battle Tournament to Ryo, who disappears to the DigiWorld right after the tournament.
2) Rika wins at least one tournament after his disappearance and becomes the Digimon Queen.
3) 10 months after the disapperance, Rika, Kazu and Kenta meet Ryo in the DigiWorld in episode 28. Although she's not obsessing with being the best anymore, Rika remembers her number one rival and the old insecurity comes crawling back. She hates Ryo for it and is really mean to him. Ryo's baffled by her behaviour but is also intrigued by her and asks Takato to say hello to her.
4) Ryo follows Rika down the chasm because he thinks she's being reckless and doesn't want her to go alone. Rika evolves to Sakuyamon and Ryo realizes how strong she actually is. He tells her how amazed he is by her.
5) Rika overcomes the rest of her obsession with being the best and realizes Ryo is a nice guy who just wants to help, she even jokes with him.
6) Ryo's refreshing personality is still annoying to her sometimes and she has no problems expressing it. However, it's not his presence that annoys her like before, it's just an occasional remark or appearance from him that can still touch a nerve.
7) Rika proves she overcame their rivalry when she gives Justimon all Sakuyamon's powers to help him defeat the Reaper.