I'm not exactly sure of what you mean by "chain" conflict, but here are some of the major conflicts in the story.
Dee is in conflict with her mother and sister because she represents materialism, while they represent the concept of having little and keeping what you do have for "everyday use."
Dee is the prettier sister: "Dee is lighter than Maggie, with nicer hair and a fuller figure." She comes home from college and is ashamed of her home, her herritage (her own name) and her family. She is trying to reach back to her African roots, which is shown by the way she changes her name, but in doing so, she is abandoning her actual heritage: "I couldn't bear it any longer, being named after the people who oppress me."
This conflict comes to a head after dinner when Dee demands that she get to have her grandma's hand-made quilts, which she wants to hang up as art. When she's told they are for her sister, she has a tantrum: "Maggie can't appreciate these quilts!" she said. "She'd probably be backward enough to put them to everyday use."
This shows that the real conflict is not just about the characters in Walker's story. It's really about whether or not old items that we cherish should be preserved as art for us to look at, or put to "everyday use." Walker clearly shows us that we can honor our heritage and our inherited belongings better by using them and passing them down.
Relvoing around a family conflict triggered by a proud, confident character Dee's desire to obtain her personal and cultural heritage but inability to appreciate genuine identity of other characters, her mother and very disparate sister, Maggie, "Everyday Use" underscores a generation gap and a contrast between two distinctively different attitudes toward heritage. Although Maggie and their mother do not attempt to understand their cultural heritiage intellectually, they know and can feel it everyday by simply living their cultural heritage, maintained int he form of family relics: the quilt.