Gasarch, Glenn and Kruskal also tried several other strategies. One promising idea leaned on randomness. A simple way to come up with a progression-free set is to put 1 in your set, then always add the next number that doesn\u2019t create an arithmetic progression. Follow this procedure until you hit the number 10, and you\u2019ll get the set {1, 2, 4, 5, 10}. But it turns out this isn\u2019t the best strategy in general. \u201cWhat if we don\u2019t start at 1?\u201d Gasarch said. \u201cIf you start at a random place, you actually do better.\u201d Researchers have no idea why randomness is so useful, he added.<\/p>\n

Calculating the finite versions of the two other new Ramsey theory results is even more vexing than determining the size of progression-free sets. Those results concern mathematical networks (called graphs) made up of nodes connected by lines called edges. The Ramsey number r<\/em>(s<\/em>, t<\/em>) is the smallest number of nodes a graph must have before it becomes impossible to avoid including either a group of s<\/em> connected nodes or t<\/em> disconnected ones. The Ramsey number is such a headache to compute that even r<\/em>(5, 5) is unknown \u2014 it\u2019s somewhere between 43 and 48.<\/p>\n