Describe the person vs person conflict between Squeaky and Gretchen. How does it start, what does it involve, and how is it resolved in end of story?

Squeaky and Gretchen are rivals. As such, there is a powerful urge in each girl to be the best in what she does so that she can feel superior to the other. But their rivalry is ended by an impressive run by Raymond.
When Squeaky and her brother Raymond "stroll" down Broadway as she does her breathing exercises, they encounter Gretchen and "her sidekicks," Mary Louise, a former friend of Squeaky's, and Rosie. 

So they are steady coming up Broadway and I see right away that it's going to be one of those Dodge City scenes....

Squeaky suggests that there is going to be a confrontation. As the girls approach Squeaky, they slow down. Squeaky is ready to fight, but Mary Louise only asks Squeaky if she is going to sign up for the May Day races. Rosie offers a challenge as she suggests that Squeaky will not win. But Squeaky will not be intimidated. She responds, "I always win cause I'm the best."
When May Day arrives, Squeaky takes her time getting to the park since the track meet is the last thing on the program for the day. When she does arrive, Squeaky looks around for her rival, Gretchen. Mr. Pearson, who pins the numbers on the competitors, taunts Squeaky, "That new girl should give you a run for your money." Then as he starts to suggest that Squeaky allow Gretchen to win, he abruptly stops and does not finish his sentence, because Squeaky gives him a threatening look. Angrily, she pins her number onto her shirt and stretches out on the grass. She sees Gretchen at the starting line. As the race begins, Squeaky passes many of the runners. She sees Gretchen with her chin out. At the same time, Squeaky notices her brother Raymond running on the outside of the fence. Suddenly, she realizes that Raymond "would make a very fine runner." This sudden thought distracts Squeaky from her intense competition with Gretchen. Instead, she describes her new mood:

And I'm smiling to beat the band cause if I've lost this race, or if me and Gretchen tied, or even if I've won, I can always retire as a runner and begin a whole new career as a coach with Raymond as my champion.

This new realization about her brother's talent distracts Squeaky from her rivalry with Gretchen. When she wins, Squeaky jumps up and down, but her elation is because of Raymond's impressive running. No longer so concerned about Gretchen, Squeaky smiles at her rival, who wins second place. Gretchen responds to Squeaky's smile by nodding to her. "We stand there with this big smile of respect between us." Their relationship improves to a real friendship.