In the first movie, Ryan shows indifference to Kelsi, but they sit at the same table in the cafeteria (as shown just before Stick to the Status Quo), and Kelsi and Ryan share a look while Sharpay oversees the cafeteria, causing some fans to already see a future couple in them.
In High School Musical 2, however, Kelsi is shown to have a locker next to Ryan's, and they presumably spent a lot of time together planning how to get Troy and Gabriella back together. In addition, Ryan and Kelsi actually speak to each other during the movies, and it is implied that Ryan supported Kelsi's ideas for the talent show before agreeing to direct the Wildcats ("Oh, hey. Kelsi's got some great new ideas to spice up the talent show..."). They are also the only characters to wear hats regularly, and their hats often match.
Ryelsi is the pairing in High School Musical 3, as Kelsi is Ryan's prom date. They sing a duet together (along with Troy and Gabriella) called Just Want To Be With You, after Ryan is ordered by Sharpay to take Kelsi to prom in order to steal Just Want To Be With You, although Ryan presummably does not do it, because of his close friendship with Kelsi.