As the title suggests, this is about Picasso and Matisse and how their competition with each other helped to spur them both to greatness. I'm not an expert on this subject so if a reader catches a mistake I hope he'll let me know about it so I can change it.
Anyway, a good place to start is this painting (above) from late in Picasso's Cubist period. He must have gotten bored with Cubism by this time because he seems to be flirting with representational painting again and with brighter color.
I think what jolted him out of Cubism was this picture (above) by Matisse. It was shockingly flat and colorful, and suggested a whole new way of depicting figures.
Picasso responded (above) by going even farther down the road Matisse had taken. Picasso flattened out his characters even more than Matisse had, and intensified and abstracted his color fields. I think he was also influenced by newspaper comics.
Actually this picture is probably from his slightly later "Heroic" period, but it still makes my point.
Matisse (above) responded to Picasso by introducing colorful patterns into his work. The whole canvas was now vibrant with pattern.
Picasso followed Matisse's background-as-pattern idea (above) and upped the ante by abstracting the background more than his rival. It didn't exactly work. Matisse's backgrounds were warm and inviting, Picasso's were clever but cold.
Interesting, eh?