A theorem that inspired this field implies that, for a party with 6 people, there must be a group of 3 mutual acquaintances or a group of 3 mutual strangers. For 10 points each:
[10m] Name this field, whose key results include the Hales-Jewett theorem. This field is sometimes thought of as an extension of combinatorics.
ANSWER: Ramsey theory
[10e] More abstractly, Ramsey's theorem asserts the existence of a minimum number of vertices that guarantees the existence of monochromatic cliques in a “complete” one of these structures.
ANSWER: graphs
[10h] This major result in Ramsey theory states that, given two integers k and p, there is some number W such that coloring the first W natural numbers with k colors yields a color with an arithmetic progression with p terms. This theorem's namesake developed a way of using dotted or undotted indices to denote right or left-handed spinors.
ANSWER: van der Waerden's theorem [or van der Waerden notation or Bartel Leendert van der Waerden]
<BB>