Rebbe suffered an amnesic illness, and had 7/13 of his teaching re-taught to him by Rabbi Chiyya and the rest by a local launderer who had overheard him studying: Nedarim 41a
Rebbe gave Rabbi Chiyya license to judge cases: Sanhedrin 5a
Rebbe sending Rabbi Chiyya to declare the new month in another town, and send a signal "King David is alive and 'standing'": Rosh HaShanah 25a
Rabbi Chiyya asked Rebbe to give judicial license to Rabbah bar Chana: Sanhedrin 5a-b
Rabbi Chiyya asked Rebbe to give judicial license to Rav; Rebbe gave him limited permission: Sanhedrin 5a-b
Rebbe insisted on accompanying Rabbi Chiyya to visit a blind Torah Scholar, and the importance of a Torah Scholar's Blessing: Chagigah 5b
Ex-communicated Rabbi Chiyya for teaching in public after Rebbe ruled that it shouldn't be done: Moed Katan 16a-b
Rabbi Chiyya pointed out Rabbi Shimon bar Rebbe's deep voice, as a blemish for a Levite; Rabbi Shimon complained to Rebbe, who pointed out Rabbi Chiyya's own speech impediment, substituing "Hey" for "Chet": Megillah 24b
Labelled Rabbi Chiyya "Iyya," an insulting alteration of his name: Moed Katan 16b
Rebbe told Rabbi Chiyya, "If you read it you didn't repeat it, if you repeated it you didn't learn it thrice, and if you did that it wasn't explained to you": Moed Katan 16b
When Rabbi Chiyya learned in Rebbe's study hall and ate at his table, he said there was no need for him to make an individual donation to the meal merging their properties to permit transportation between them on Shabbat. Rav did the same, when he learned in Rabbi Chiyya's study hall: Eruvin 73a
Rav Ashi said, regarding a lesson taught by both Rebbe and Rabbi Chiyya, "Had Rebbe not taught it, how would Rabbi Chiyya have known it?": Eruvin 92a
Rebbe wanted to marry his son to Rabbi Chiyya's daughter. She died before it could be completed, and they examined their lineages and realized that Rebbe's family, descendants of King David, could not marry Rabbi Chiyya's family, descendants of King David's brother Shimi: Ketuvot 62b