In Toni Cade Bambara's short story, "Raymond's Run," Squeaky, the narrator of the story, is serious about her running. Gretchen is her main competitor. The conflict between them centers around their competition during the May Day races.
Squeaky describes this conflict as she is giving information about her prowess as a runner. She states that no one can beat her, except for her father. Shortly after that statement, she says,
I’m the fastest and that goes for Gretchen, too, who has put out the tale that she is going to win the first-place medal this year. Ridiculous. In the second place, she’s got short legs. In the third place, she’s got freckles. In the first place, no one can beat me and that’s all there is to it.
One day when Squeaky is walking in the neighborhood with her brother Raymond, she encounters Gretchen and her friends. They strike up a conversation about the May Day races, and one of Gretchen's friends even suggests that Squeaky won't win this year, which heightens the conflict between Squeaky and her greatest rival.
So I’m strolling down Broadway breathing out and breathing in on counts of seven, which is my lucky number, and here comes Gretchen and her sidekicks: Mary Louise, who used to be a friend of mine when she first moved to Harlem from Baltimore and got beat up by everybody till I took up for her on account of her mother and my mother used to sing in the same choir when they were young girls, but people ain’t grateful, so now she hangs out with the new girl Gretchen and talks about me like a dog; and Rosie, who is as fat as I am skinny and has a big mouth where Raymond is concerned and is too stupid to know that there is not a big deal of difference between herself and Raymond and that she can’t afford to throw stones. So they are steady coming up Broadway and I see right away that it’s going to be one of those Dodge City scenes cause the street ain’t that big and they’re close to the buildings just as we are. . . I’m ready to fight, cause like I said I don’t feature a whole lot of chit-chat, I much prefer to just knock you down right from the jump and save everybody a lotta precious time. "You signing up for the May Day races?" smiles Mary Louise, only it’s not a smile at all.
Squeaky only addresses Gretchen in this exchange and lets her know that she has every intention of not only signing up but winning the May Day races like she always does.
After the race, the two girls show respect for each other by smiling, and as Squeaky describes it, it is about the best smile two girls can do for each other. This doesn't prove the conflict between them is resolved, but they have gained each other's respect.